What to Say during Tawaf
Question

These adhkar were compiled by someone who loves ‘umrah and wanted to distribute them among those who perform ‘umrah, but he is waiting until he can consult you. Please explain which are sound and which are not. 
Some of them are adhkar that the pilgrim needs, such as what is to be said during tawaf, starting the first circuit with praise of Allah, then sending blessings upon His Prophet, then du‘a (supplication), giving precedence to supplication concerning religious matters over worldly matters, with proper focus and presence of mind.
Praise be to Allah.As far as we know, there is no report from the Prophet (blessings and peace of Allah be upon him), concerning du‘as and adhkar to be recited in tawaf, apart from what is to be said between the Yemeni Corner and the Black Stone : “Rabbana atina fi’d-dunya hasanah wa fi’l-akhirati hasanah wa qinna ‘adhab an-nar (Our Lord, give us in this world [that which is] good and in the Hereafter [that which is] good and protect us from the punishment of the Fire)” [narrated by Ahmad in al-Musnad (3/411); classed as sahih by Ibn Hibban (9/134), al-Hakim (1/625)]; and takbir (“Allahu akbar”) every time one comes in line with the Black Stone [narrated by al-Bukhari (4987)].
With regard to the remainder of tawaf, the pilgrim has the choice between dhikr, du‘a and reciting Quran.
Ibn Qudamah said in al-Mughni (3/187):
“It is mustahabb to offer supplication (du‘a) and remember Allah, may He be exalted, a great deal (dhikr) whilst doing tawaf, because that is mustahabb in all circumstances, so in the case of this act of worship it is even more apt. It is also mustahabb to refrain from speaking, apart from remembering Allah, may He be exalted, or reciting Quran, or enjoining what is right and forbidding what is wrong, or saying what cannot be avoided.” 
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said in Majmu‘ al-Fatawa (26/122):
“In tawaf, there is no specific dhikr that was narrated from the Prophet (blessings and peace of Allah be upon him), that he either enjoined, said or taught. Rather whilst doing tawaf, the pilgrim may offer any Islamically-prescribed du‘a. What many people mention of a specific du‘a to be recited beneath the mizab (downspout) and the like, has no basis. 
The Prophet (blessings and peace of Allah be upon him) used to end his tawaf between the two corners [the Yemeni Corner and the Black Stone corner] with the words, “Rabbana atina fi’d-dunya hasanah wa fi’l-akhirati hasanah wa qinna ‘adhab an-nar (Our Lord, give us in this world [that which is] good and in the Hereafter [that which is] good and protect us from the punishment of the Fire)”, as he used to conclude all his du‘as in that manner. There is no specific dhikr required in tawaf, according to the consensus of the leading scholars.”
Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) said:
“It is proven that the Prophet (blessings and peace of Allah be upon him) used to say takbir (“Allahu akbar”) every time he came to the Black Stone, and between the Yemeni Corner and the Black Stone, he used to say “Rabbana atina fi’d-dunya hasanah wa fi’l-akhirati hasanah wa qinna ‘adhab an-nar (Our Lord, give us in this world [that which is] good and in the Hereafter [that which is] good and protect us from the punishment of the Fire)” [al-Baqarah 2:201].
There is no report from the Prophet (blessings and peace of Allah be upon him) to suggest that there is a specific du‘a to be recited in each circuit of tawaf.”
Based on that, the one who is circumambulating the Ka‘bah may offer whatever supplication he likes, asking Allah for what is good in this world and the Hereafter, and remembering Allah, may He be exalted, by reciting any prescribed dhikr, such as saying Subhaan Allah (glory be to Allah), al-hamdu Lillah (praise be to Allah), La ilaha ill-Allah (there is no God but Allah), and Allahu akbar (Allah is most great), or reciting Quran.” (Majmu‘ al-Fatawa, 24/327)
For more about tawaf, please see these answers:109246 , 234172 , 106598 , 34644 , and 253569
And Allah knows best.