It has been reported from Muawiyah bin Ammar, quoting Abi Abdillah (a.s.), as saying that, during tawaf, the following supplication be said :

اَللّٰہُمَّ اِنِّیْ اَسْئَلُکَ بِاسْمِکَ الَّذِیْ یُمْشٰی بِہٖ

Allahumma innii asaluka bismikallazii yumshaa bihi

O Allah! I beseech You by Your name which makes possible motion

عَلٰی طَلَلِ الْمَآءِ کَمَا یُمْشٰی بِہٖ عَلٰی جُدَدِ الْاَرْضِ

alaa zulalil maai kama yumshaa bihi alaa judadil arzi

in the darkness of water as it does on land.

وَ اَسْئَلُکَ بِاسْمِکَ الَّذِیْ یَہْتَزُّ لَہٗ عَرْشُکَ

wasaluka bismikal lazii tahtazuu lahu arshoka

I beseech You by Your Name which makes Your Arsh (Throne) swing.

وَ اَسْئَلُکَ بِاسْمِکَ الَّذِیْ تَہْتَزُّ لَہٗ اَقْدَامُ مَلَآءِکَتِکَ

wasaluka bismikal lazii tahtazuu lahu aqdaamu malaikatika

I beseech You by Your Name which puts in motion the feet of Your angels.

وَ اَسْئَلُکَ بِاسْمِکَ الَّذِیْ دَعَاکَ بِہٖ مُوْسٰی

waasaluku bismikallzii daaka bihi Musa

I beseech You by Your name by which Moosa (Moses)

مِنْ جَانِبِ الطُّوْرِ فَاسْتَجَبْتَ لَہٗ

min janibi-tturi fastajabta lahu

supplicated to You on Mount Sinai

وَ اَلْقَیْتَ عَلَیْہِ مَحَبَّۃً مِّنْکَ

waaalqaita alaihi mahabbatan minka

and You granted his supplication and met him with love.

وَ اَسْئَلُکَ بِاسْمِکَ الَّذِیْ غَفَرْتَ بِہٖ

waasaluka bismikallazii ghafarta bihi

I beseech You by Your name by which You granted immunity from sins

لِمُحَمَّدٍ صَلَّی اللّٰہُ عَلَیْہِ وَ آلِہٖ

li-muhammadin sallallahu alaihi waalihi

to Muhammad Peace be upon him and his progeny

مَا تَقَدَّمَ مِنْ ذَنْبِہٖ وَ مَا تَاَخَّرَ

maa taqaddama min dhanbihi wamaa taakhara

and completed Your bounty on him,

وَ اَتْمَمْتَ عَلَیْہِ نِعْمَتَکَ اَنْ تَفْعَلَ بِیْ کَذَا وَ کَذَا

waatmamta alaihi niimataka an tafal bii.

treat me with (ask for your wishes).