Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said in Majmu’ al-Fatawa (25/290):
“As for fasting in Rajab in particular, the ahadith concerning that are all da’if (weak), and in fact mawdu’ (fabricated). The scholars do not rely on any of them. They are not among the da’if ahadith which have been narrated concerning virtues, rather most of them are fabricated and false. In al-Musnad and elsewhere there is a hadith which says that the Prophet (peace and blessings of Allah be upon him) enjoined fasting the sacred months, namely Rajab, Dhu’l-Qa’dah, Dhu’l-Hijjah and Muharram, but this has to do with fasting during all of them, not just Rajab.”
Ibn al-Qayyim (may Allah have mercy on him) said:
“Every hadith which mentions fasting in Rajab and praying during some of its nights is false and fabricated.” (Al-Manar al-Munif, p. 96)
Al-Hafiz ibn Hajar said in Tabyin al-‘Ajab (p. 11)
“There is no sahih hadith that would count as evidence which speaks of the virtue of the month of Rajab, or that speaks of fasting this month or part of it, or of spending any particular night of it in prayer.”