Abdullah bin Umar used to walk briskly during the first three circuits and walk normally during the remaining four, and he claimed that the Messenger of Allah ﷺ used to do this.
أخبرنا عبيد الله بن سعيد قال حدثنا يحيى عن عبيد الله
أن عبد الله بن عمر كان يرمل الثلاث ويمشي الأربع ويزعم أن رسول الله ﷺ كان يفعل ذلك
Bukhari narrated it in the Book of Hajj, Chapter: Whoever performs tawaf of the House upon arriving in Makkah, etc., hadith no. 1617, etc., hadith no. 1261/230, and Muslim, Book of Hajj, Chapter: The recommendation of hastening (raml) during tawaf in Umrah, with different wordings but the same meaning, and it is also in al-Kubra, hadith no. 3938, the hadith of Ubaydullah bin Umar, with this chain.