Besides, party General Secretary K.C. Venugopal has also declined to accept the post of Working President offered to him, the sources said.
Senior party leaders Ahmed Patel and Ghulam Nabi Azad, who have been tasked to look for a face outside the Gandhi family to lead the party, had asked Antony to take up the job.
The sources said Antony informed Patel and Azad that he has full respect for the Gandhi family but could not accept the post.
Venugopal refused the offer, citing his other role of strengthening the party.
He is also in-charge of Karnataka, where the party won only one out of 28 seats in the Lok Sabha elections, despite being in power in the state along with the JD-S after bagging 78 out of 225 seats in the Assembly polls last year.
The search for the new leader was initiated after Rahul Gandhi decided to resign from the party's top post, following the dismal performance in the recently-held general elections.
Although the Congress Working Committee, the party's highest decision-making body, had rejected Gandhi's resignation, he has remained steadfast on giving up the post.
Now Congress leaders are looking for a face from north India to lead the party, the party source said.
Names of several senior leaders are doing the round for the role of Working President.
The party has considered the Presidium system as well as the option of having multiple Working Presidents.