Bathing on Erev Shabbos during the nine days:^[1] 

Erev Shabbos Chazon:^[2] On Erev Shabbos Chazon [Parshas Devarim] one may bathe his hands, feet and head in cold water.^[3] Some allow one who bathes himself every Erev Shabbos, to wash his head [hands and feet^[4]] even in hot water.^[5] It is forbidden to wash the remainder of one’s body even with cold water.^[6] [Likewise, one may not use soap or shampoo to wash his head, hands or face.^[7] The above however only applies when bathing for purposes of pleasure. It is however permitted to bathe the body for reasons of cleanliness, as stated above in C, and so is an old age Chassidic custom to allow bathing the body [for cleanliness purposes] on Erev Shabbos Chazon.^[8] Accordingly, if one feels sweaty or dirty, he may shower even with hot water, until his body is clean. It is however forbidden to remain in the water for more time than necessary.^[9]]

Rosh Chodesh Menachem Av falls on Erev Shabbos:^[10] When Rosh Chodesh Menachem Av falls out on Erev Shabbos then one who is careful to bathe every Erev Shabbos is permitted also on this Erev Shabbos to bathe his entire body in hot water as usual.^[11] Furthermore, it is forbidden to be stringent in this matter.^[12] [One may likewise immerse in a hot Mikveh this Erev Shabbos.^[13]]

[1] Rama 551:16

[2] Rama ibid

Letter of law or custom: Ideally, it should be permitted to bathe on Erev Shabbos due to Kavod Shabbos, just as we permit laundering clothing in honor of Shabbos. However, it is accustomed to prohibit bathing even on Erev Shabbos. [Darkei Moshe 551; M”A 551:14] 

[3] Rama ibid; Maharil 17

[4] Chayeh Adam 133:19; brought in M”B 551:97; Kitzur SHU”A 122:13; Kaf Hachaim 551:201

Other opinions: The Rama ibid rules one may wash these parts only with cold water. 

[5] Rama ibid

[6] Rama ibid; Darkei Moshe 551:7; M”A 551:41; Elya Raba 551:37

[7] M”A 551:41; Chayeh Adam 133:19; M”B 551:97; Kitzur SHU”A 122:13; Kaf Hachaim 551:200

[8] Mishmeres Shalom Kudinov 40:1; Rav Henkin that so is custom and so was custom of Chassidei Reisin; See Nitei Gavriel 43:2

The reason: As it is permitted to bathe for the sake of a Mitzvah, as well, according to Kabala, one is not to do any change between this Shabbos and other Shabbosim. [Mishmeres Shalom ibid]

[9] See Tosefes Chaim ibid; Piskeiy Teshuvos 551:48; Rav Eli Landa

[10] M”A 551:40 in name of Bach; Taz 551:16; Elya Raba 551:36; Rashal 92; first custom in Levush, brought in M”A ibid; Bach 551, brought in M”A and Taz ibid “The custom is not to forbid it and one who avoids bathing this Erev Shabbos Lekavod Shabbos is doing a sin”; Ateres Zekeinim 551; Derech Hachaim 8; Kitzur SHU”A 122:13; M”B 551:89; Kaf Hachaim 551:189

Other customs: Some Poskim rule it is forbidden to bathe this Erev Shabbos just like on Erev Shabbos Chazon, being it is the first day of the nine days. [second custom in Levush ibid; Peri Megadim 551 M”Z 16 “I have not seen people be lenient in this”, brought in Shaareiy Teshuvah 551:17]

[11] One who is not careful to bathe every Erev Shabbos: The Taz ibid implies that if one is not particular to bathe every Erev Shabbos then he may not bathe this Erev Shabbos. [Taz ibid concludes that only one who is careful to bathe every Erev Shabbos may be lenient like the Bach] However the Bach implies it is permitted even for one who is not particular and if one is particular then it is even a sin.

[12] Bach ibid, brought in Taz and M”A ibid

[13] Piskeiy Teshuvos 551:51