Jaishankar, 64, had joined the BJP on Monday evening. The career bureaucrat served as the Foreign Secretary before he was inducted into the Union Cabinet by Prime Minister Narendra Modi, who entered his second term last month.
The two seats, for which Jaishankar and Thakore filed nominations, were vacated after BJP leaders Amit Shah and Smriti Irani were elected to the Lok Sabha on May 23.
Thakor is from North Gujarat's Mehsana district, to which Narendra Modi also belongs, and he represents the dominant Thakor OBC community in the state.
Chief Minister Vijay Rupani and Gujarat BJP President Jitu Vaghani were present when Jaishankar and Thakore submitted their nomination forms to Returning Officer C.B. Pandya.
The last date of submitting nomination papers is June 25, while scrutiny of papers would be conducted on June 26, and the last date of withdrawal of names is June 28.
Given its strength of 100 MLAs in the Assembly, the BJP is set to win both the seats, since the elections for both the seats will be held separately according to the Election Commission (EC) notification.
The Congress had challenged the EC decision to hold separate polls for the upper House seats in the Supreme Court and demanded that the elections should be held together, as was the practice.
Given Congress strength of 71 MLAs in the Assembly, the Congress and the ruling BJP would have won one seat each on the basis of first preference votes of their MLAs had the polls been held together. In the case of separate polls, all legislators vote twice and first preference votes are counted for each seat.
The Congress has fielded Gaurav Pandya and Chandrika Chudasama for the two Rajya Sabha seats. Leader of Opposition Paresh Dhanani said the Congress' petition was being heard in the Supreme Court but the party had fielded candidates on both the seats in continuation of the election process since Tuesday was the last day to file nominations.
"We will pursue our petition in the Supreme Court till its logical end and fight it out in the larger interest of democracy," Dhanani told reporters.
( With inputs from IANS )