Three Plaid Cymru AMs have signed a letter calling for a challenge to Leanne Wood's leadership, BBC Wales has learned.
Llyr Gruffydd, Sian Gwenllian and Elin Jones have asked colleagues to consider putting their names forward.
The window which allows a leadership challenge closes on 4 July.
In a BBC Wales interview Ms Wood said she would stand down after the assembly election in 2021 if it did not result in her becoming first minister.
The Plaid Cymru group on Carmarthenshire council has backed Adam Price to stand for the leadership.
Cllr Alun Lenny told BBC Radio Cymru's Taro'r Post programme the group had asked Mr Price if he would be willing for his name to be put forward officially.
Mr Price has yet to respond, Mr Lenny said.
Another senior Plaid AM, Rhun ap Iorwerth, has insisted he has "no plans" to challenge for the leadership and told journalists that he wanted Ms Wood to lead the party into the next election.
Sources close to both Mr Price and Mr ap Iorwerth have told BBC Wales they are coming under increasing pressure to challenge Ms Wood.
Two Plaid Cymru constituency parties are meeting in the next 10 days to discuss leadership nominations.
It is expected they will agree to put Mr Price's name forward.
Plaid Cymru has declined to comment.
Sources say party branches from across Carmarthenshire had written to Mr Price urging him to stand.