Nafi' (may Allah have mercy on him) says that Sayyiduna Ibn Umar (may Allah be pleased with them both) used to perform ramal (brisk walking) in the first three circuits of tawaf from the Black Stone to the Black Stone, and in the remaining four circuits he would walk at a normal pace, and he used to state that the Prophet (peace and blessings of Allah be upon him) did likewise.
Hadith Referenceمسند احمد / مسند المكثرين من الصحابة / 5943
Hadith Gradingحکم دارالسلام:حديث صحيح، عبد الله العمري - وإن كان ضعيفا- متابع.