It is sunnah to offer a two rak'ah prayer after completing the tawaf at the Station of Ibrahim (peace be upon him), or at any other place in the Sacred Mosque, regardless of whether it is a supererogatory tawaf or obligatory one (as in Hajj or 'Umrah).
Jabir reported that "When the Prophet (peace be upon him) came to Makkah, he went around the Ka'bah seven times, then went to the Station of Ibrahim and recited the verse (Qur'an 2.125): "And take of the Station of Abraham a place of prayer." He prayed behind it, and then went to the Black Stone and kissed it." (Reported by Tirmizhi who considers it a sound hadith)
In this two rak'ah prayer, it is sunnah to recite Surah Al-Kafirun, in the first rak 'ah, and Surah Al-Ikhlas in the second rak 'ah, after reciting Al-Fatihah. (Reported by Muslim and others) These two rak'ahs may be offered at any time of the day and night including the prohibited times.
Jubair bin Mut'im reported that the Prophet (peace be upon him) said, "O Bani Abd Manaf! Do not prevent anyone from performing tawaf around the Ka'bah. One may pray (in the Sacred Mosque) any time during the day or night." (Reported by Ahmad, Abu Dawud, and Tirmizhi who regards it a sound hadith) Ash-Shafi'i and Ahmad hold this view.
Though it is sunnah to offer this two rak'ah prayer after the tawaf in the Sacred Mosque, it may also be offered outside the Mosque. Umm Salamah reported that she performed a tawaf around the House riding, but she did not offer the two rak'ah prayer until she had left the Mosque. (Bukhari) Malik reported from 'Umar that he offered these two rak'ahs at Zhi Tuwa valley. Bukhari also has reported that 'Umar prayed (these two rak'ahs) outside the Sacred Mosque.
If one offered a prescribed prayer after the tawaf then he need not offer this two rak'ah prayer. The Shafi'i school holds this view, which is also in accordance with the well known position of Ahmad. The Maliki and the Hanafi schools are of the opinion that these two rak'ahs must be offered and no other prayer can replace them.