Praise be to Allah.
Yes, it is permissible for a person to take a bath naked, because this was reported from the Prophet (peace and blessings of Allaah be upon him) as was reported by Maymoonah (may Allaah be pleased with her(, w ho said that she and the Prophet (peace and blessings of Allaah be upon him) took a bath together, washing from a large bowl in which there were traces of dough. And it was reported in several places as narrated by al-Bukhaari and Muslim, and he bathed with Aaishah (may Allaah be pleased with her).
There is no saheeh hadeeth from the Prophet (peace and blessings of Allaah be upon him) to indicate that it is obligatory to cover oneself when bathing. It was reported that some of the Salaf used to cover themselves, but this is not obligatory according to Islam. It is sufficient to be far from where people can see you.
Some scholars use as evidence the hadeeth in al-Saheehayn which speaks of Moosa (peace be upon him) bathing naked (al-Bukhaari, 274; Muslim, 339). Al-Nawawi entitled the chapter in Saheeh Muslim in which this hadeeth occurs the permissiblity of bathing naked when one is alone.
It was also reported that Ayyoob (peace be upon him) bathed naked. (Saheeh al-Bukhaari, 275).
Imaam al-Nawawi said:
We have already seen in the previous chapter that it is permissible to uncover the awrah when necessary, when one is alone, such as when taking a bath, urinating, engaging in intimate relations with one's wife, and so on. In all of these cases it is permissible to uncover ones awrah when one is alone, but in front of other people it is forbidden to uncover the awrah in all cases.
The scholars said:
Covering oneself with a wrapper or something similar when taking a bath alone is preferable than uncovering oneself. Uncovering oneself is permissible for the period of time that is essential to take a bath, etc., but uncovering oneself for longer than that is haraam according to the most correct opinion and Allaah knows best.