This is barely a week after President Robert Mugabe and his Zanu-PF party were declared victors in the Zimbabwe elections.
Members of the MDC-T's top decision-making body, the national council, said there was growing frustration within their ranks over Tsvangirai's leadership.
Although the members, who preferred to remain anonymous, said they would not immediately move for a motion of no confidence in Tsvangirai, they said there were plans to dethrone him at the party's next congress in 2016 if he did not step down voluntarily.
Although the MDC-T officials were unanimous in their conviction that Zanu-PF had rigged the election, there is unhappiness that the rigging happened right under their leaders' noses, especially Tsvangirai's, who they say "appeared to have a cosy relationship with Mugabe".
A senior party official and former government minister said the relationship between Mugabe and Tsvangirai worried them and he believed Mugabe had used this closeness to "dribble past Tsvangirai".
"There is no doubt that this dictatorship won a dirty election, but some in the leadership were so gullible and naive. Tsvangirai should come clean and tell the nation what role he played in this farce. Mugabe pulled him so close that he began trusting him against our better advice," he said.
"He is joining us in crying 'rigging', when he was the first to defend the Zimbabwe Electoral Commission [ZEC]. Did he not defend the ZEC in public when some of us were calling for the removal of the secretariat, which is littered with intelligence operatives and the military?"
There was concern about the appointment of Judge Rita Makarau to head the ZEC in light of her lack of independence and links to Zanu-PF, but Tsvangirai defended the appointment at the time, saying "everyone has a past".
Another top official said the party was also unhappy because Tsvangirai – who "did not think strategically" – had joined Mugabe in sidelining the president of the smaller MDC, Welshman Ncube, in favour of outgoing deputy Prime Minister Arthur Mutambara. The two men were locked in a battle over leadership.
"He connived with Mugabe and Mutambara to hijack the constitution-making exercise from Parliament, trashing the party's national council resolution in the process, resulting in Mugabe having his way."
The official said that, although Ncube had not won any seats in Parliament, a united front would have boosted the image of Tsvangirai's party.
"He [Tsvangirai] is the same man who spent time trying to settle personal scores with Ncube and sidelining him from that cursed principals' forum, against Southern Africa Development Community resolutions, only to seek an alliance with Ncube on the eve of the election."
The official said that although the combined vote of the two MDCs would not have been enough to beat Zanu-PF, given the results announced by the ZEC, a grand alliance could have given people more hope, translating to bigger voter participation and making it more difficult for the poll to be rigged.
MDC-T spokesperson Douglas Mwonzora said the party was still behind Tsvangirai and defended its unanimous decision to go into the flawed elections. It had contested the elections to defend the constitution "which Zanu-PF wanted to change".
"There are some attempts by certain individuals to divide the party using the press. But, as things stand, we are solidly behind Morgan Tsvangirai. We have no doubt that he won the election and was defrauded," said Mwonzora.
He added that all positions in the party would be up for grabs at its 2016 congress and people were free to challenge any incumbent.