These verses are mistranslated in pretty much every English Bible commonly available. Both verses are printed below in Greek, then transliterated, and then with correct translations, and explanations about the translations.
1 Corinthians 6:9
Ἤ οὐκ οἴδατε ὅτι ἄδικοι Θεοῦ βασιλείαν οὐ κληρονομήσουσιν; Μὴ πλανᾶσθε· οὔτε πόρνοι οὔτε εἰδωλολάτραι οὔτε μοιχοὶ οὔτε μαλακοὶ οὔτε ἀρσενοκοῖται...
I ouk idhate oti adhiki Theou vasilian ou klironomisousin? Mi planasthe; oute porni oute idhololatrai oute mikhi oute malaki oute arsenokitai...
(Transliteration of Modern Greek pronunciation.)
Or haven’t you known that the unjust will not inherit the kingdom of God? Do not be misled; neither fornicators nor idolaters nor adulterers nor soft ones* nor those who lie with males...**
1 Timothy 1:10
πόρωοις, ἀρσενοκοίταις, ἀνδραποδισταῖς, ψεύσταις, ἐπιόρκοις, καὶ εἴ τι ἕτερον τῇ ὑγιαινούσῃ διδασκαλίᾳ ἀντίκειται,
…pornis, arsenokitais, andhrapodhistais, psevstais, epiorkis, kei i ti eterov ti iyi-einousi dhidhaskalia antikitai,
(Transliteration of Modern Greek pronunciation.)
…to fornicators, to those who lie with males,** to kidnappers, to liars, to perjurers, and if any other thing opposes healthful teaching,
*Soft ones: The Greek word malaki (mala-KEE) is a plural noun, derived from the adjective malakos (mala-KOS). The adjective means infirm (from illness) or soft or fine, but in meaning soft or fine, is restricted to describing material or clothing. It describes the type of clothing worn by wealthy people. This adjective was used in Luke 7:25, when Jesus asked the crowd if they had gone out to the wilderness expecting to see someone dressed in fine clothing. Coining a noun from this adjective would logically suggest the type of people who wear soft or fine clothing. Jesus Himself indicated the near impossibility of wealthy people entering the kingdom.
**Those who lie with males: The Greek word arsenokitai (arseno-KEE-tay) (the form used in 1 Timothy is arsenokitais [arseno-KEE-tays]), is formed by combining the noun arsin, which means male, with the construction kit-, a derivative of the verb kimei, which means lie down. Combined, the word refers to people who lie down with males. What remains to be determined is whether the word is referring to males lying with males, or females lying with males. Ordinarily, to determine if a Greek noun is masculine or feminine, one looks at it in the nominative case with the definite article. For example, o adhelfos, the brother, is in the nominative case, and both the os ending and the definite article o tell us the noun is masculine. But the word used in these two verses presents a small challenge to us, because in ancient literature, it never appears with a definite article. (In fact, outside these two passages, it never appears at all!) Of course, we could simply look it up in a Modern Greek dictionary, and it would tell us the word is masculine and means homosexual. But is that the end of it? Actually, no. The dictionary’s definition and assignment of gender is based on centuries of preconceived notions about what Paul was saying, and not on actual usage from the first century. So in this case, the dictionary can’t answer the question for us. We need to look back to the word itself, and its context, to search for clues.
The last two letters of the word in 1 Corinthians, and the last three in 1 Timothy, are where we need to look first. Greek nouns are declined according to case. That is, the ending of a noun changes to indicate how the word is being used in the sentence. We have something similar in English pronouns: We use the word I as a subject, but me as an object. For all intents and purposes, I and me mean the same thing. But it is incorrect to say Me want a book, or Give I a book. In 1 Corinthians, the word is in the nominative case, and the ending is one that is often feminine. This would suggest that the word is referring to women lying with males. In 1 Timothy, the word is in the dative case, which in English corresponds to putting the word to before the noun. (Example: Give the book TO ME.) And again, the ending is one that is often feminine. The fact that this word is not found anywhere else in literature of the period suggests that Paul himself coined it. Although there was no such word as homosexual at the time, there were words that could be used to indicate sexual activity between persons of the same sex. Had Paul intended to refer to homosexuals here, common sense would have him use words people already knew and understood. But the creation of a new word suggests a different concept. Had he intended for his new word to be understood to refer to males, he would have given it a different ending. The plural ending oi (ois in the dative case) is never feminine, and would have served the purpose. But he didn't do that. So even without the definite article to prove the point, the evidence so far suggests that Paul was speaking about women when he used this word, not men.
There are two other pieces of evidence to weigh. First, when properly translated, Scripture contains no prior condemnation of homosexuality, and the Hebrew Old Testament contains the record of two same-sex marriages, neither condemned by God. Paul, as a Jewish scholar, could not have been ignorant of this. So for him to suddenly, and without precedent, introduce a condemnation of homosexuality, without a word of explanation, would make absolutely no sense, and would have created an uproar in the early churches. Church history documents that same-sex marriages existed, and continued, in the Christian church up until around the 13th or 14th century.[i]
Another point we should note is context. There are certain other words in these verses that refer to sexual misconduct. These are porni and mikhi, fornicators and adulterers. Both of these words are masculine. Although masculine plurals in Greek can refer to women, if there are men included also, there is a more sociological reason for these words being masculine. In the mindset of the first century, heterosexual relations were always thought of in terms of the male being the active partner, and the female being passive. That is, the man was thought to be the initiator of sexual acts. Therefore, the words describing these things tend to be masculine: Adulterer, rather than adulteress. And it could be argued by some that these words applied only to men, and didn’t apply to women at all. Grammatically, they could be understood that way. But certainly, few would suggest that Paul was opposed to sexual misconduct by men, but completely ambivalent to such activity by women. And few would subscribe to the notion that sexual misconduct was always initiated by males, and never by females. Even Paul knew that wasn’t the case. The word arsenokitai would address this: There are specific situations where a female initiates sexual activity. For example, a prostitute usually approaches a prospective customer, soliciting him. So this word would deal with situations where a female initiates sexual misconduct.
The word was only used twice in the first century, both times by Paul, and not used again until the second century. In the second century, it was used to mean female prostitutes. Another reason for Paul to create a new word for prostitutes was that he had already used the original word to mean something else. Centuries earlier, in earliest Greek, the porn- root referred only to prostitutes and prostitution. But over the centuries, it had expanded in meaning, and now referred to any extra-marital sex. Paul had already used the word to mean fornicators. Since prostitution was both a business and a form of worship of the fertility goddess, some could have claimed that Paul didn't have actual prostitutes in mind when he said fornicators. Again, this word covers a loophole. Historically, only relatively recently have people begun to think this word refers to homosexuals, and newer translations actually render the word that way, despite the lack of grammatical or historical support.
[i] “Same-Sex Unions in Premodern Europe” by John Boswell.