The short answer is maybe, but it appears unlikely.
To become a U.S. citizen, your husband would need to first become a Lawful Permanent Resident ("Green Card" holder). If he has a Green Card for three years and remains married to you and lives with you for three years, he would be eligible to obtain U.S. Citizenship.
Therefore, the real issue here is whether he can obtain a Green Card. It appears that your husband had falsely claimed to be a U.S. citizen. Falsely claiming to be a U.S. citizen bars someone from obtaining a Green Card through marriage to a U.S. citizen. There are other avenues of relief for which he may be eligible, but that would require more information.
Finally, if your husband is the son of a U.S. citizen or otherwise had a reasonable belief that he was indeed a U.S. citizen at the time he claimed to be one, there may be a possibility that he can have the permanent bar waived, but it does not seem from what you have stated that he had such a reasonable belief.