In the name of Allaah, the most Gracious, the most merciful.
No! Because, there is no any Soorah in the Qur'an or any Saheeh hadeeth to indicate that there is any special virtue to in fasting all or part of this month and we heard that Prophet Muhammad said:
The best of speech is the Book of Allaah, and the best of guidace is the guidance of Muhammad, the evil of thing are those which are newly invented(in religion) and every innovation is going astray"[Reported by Muslim and al-Nisaai].
There is no Saheeh hadeeth to indicate that there is any special virtue in taking the month of Rajab as fasting month.
What some people do, singly out some days of Rajab for fasting, believing that they are better than others, has no basis in Sharee'ah.
But there is a report from the Prophet(PBUH) which indicates that it is mustahaabb to fast during the sacred months(and Rajab is the part of them). The prophet(PBUH) said:
"fast somedays of the sacred months and not others" Narrated by Abu Dawood, 2428. Clased as da'eef by Al-Albaani in Da'eef Abi-Dawood.
Even if this hadeeth were saheeh, it is mustahaabb to fast during Rajab because of this, and he also fast in the other sacred months, there is nothing wrong with that. But singling out of Rajab for fasting is not right.
Shaykh al-Islam Ibn Taymiyah(may Allaah have mercy on him) said in majmu' al-fatawa(25/290).
As for fasting in Rajab in particular, the ahaadeeth concerning that are all da'eef(weak) and in fact mawdoo'(fabricated).
The scholars do not rely on any of them, the are not among the da'eef ahaadeeth which have been narrated concerning virtue, rather most of them are fabricated and false. In Musnad and elsewhere there is a hadeeth which says that the Prophet(PBUH) enjoined fasting the sacred months namely Rajab, Dhul-Qadah, Dhul-Hijjah and Muharraam, but this has to do with fasting during all of them, not just Rajab.
"Ibn al-Qayyim(may Allaah have mercy on him) said: Every hadeeth which mentions fasting in Rajab and praying during some of it night is false and fabricated"[al-Manaar al-Muneef, p.96].
Al-Haafiz Ibn Hajar said in Taybeen al-'Ajab(p. 11).
There is no saheeh hadeeth that would count as evidence which speaks of the virtue of the month of Rajab, or that speak of fasting of this month or part of it, or of spendind any particular night of it in prayer.
Shaykh Sayyid Saabiq(may Allaah have mercy on him)said in Fiqh al-Sunnah(1/282):
"Fasting in Rajab is no better than fasting in any other month, exept that if is one of sacred months. There is no report in the Saheeh Sunnah to suggest that there is anything special about fasting in this month. Whatever has been narrated concerning that is not fit to be quoted as evidence.
Shaykh Ibn Uthaymeen(may Allaah have mercy on him) was asked about fasting on the twenty-seventh of Rajab and spending that night in prayer. He replied:
Fasting on the twenty seventh of Rajab and spending that night in prayer is a bid'ah(innovation) and every bid'ah is going astray.
Prophet Muhammad(PBUH) said:
"Whoever does any deed that is not part of this matter of ours will have it rejected"[Muslim].
And he also said: Stick to my Sunnah and the Sunnah of any rightly guided caliph, beware of newly invented matter, for every newly invented matter is Bid'ah and every Bid'ah misleading or astrayed.
Note: If you are fasting the Sunnah fasting(Thursday, monday of the weak, 13th, 14th and 15th of the month) before, you can continue it in Rajab.