Mr Newton was criticised by MLAs after he allowed his party leader, First Minister Arlene Foster, to make a statement on the cash-for-ash scandal.
She addressed MLAs without approval of Deputy First Minister Martin McGuinness, from Sinn Féin.
Mr Newton has faced calls to step down as speaker from the UUP and SDLP.
Opposition politicians staged a walk-out protest on Monday when Mrs Foster began to make her statement on the botched Renewable Heat Incentive (RHI) scheme.
They complained that the speaker had undermined the principles of power-sharing by permitting the first minister to make a formal statement to the chamber without the agreement of the deputy first minister.
On Thursday, Sinn Féin met the Alliance Party to seek their support for an Assembly motion that will call on Mrs Foster to step down temporarily while the RHI scheme is investigated.
After the meeting, former Sinn Féin minister Carál Ní Chuilín said: "We also discussed the position of the Speaker following Monday's shambolic and partisan proceedings in the Assembly.
"He has undermined the integrity and the impartiality of the Speaker's office and Sinn Féin believes his position is untenable."
Earlier this week, Mr Newton released a letter he had sent to MLAs acknowledging what he said were genuine frustrations and concerns over the nature of Monday's proceedings.
In response to calls for him to step down, Mr Newton said he would reply in writing to opposition MLAs in January.
His party leader set up the RHI scheme in 2012 when she was minister for enterprise, trade and investment, and she has faced intense criticism for not imposing cost controls.
As a result, the scheme has gone vastly over budget and has landed the public purse with an unexpected £400m bill, which may have to come out of Stormont's block grant.
Mrs Foster last week apologised for the lack of cost controls and said she was working on a plan to reduce the overspend.
The position of the Assembly Speaker Robin Newton has become "untenable" as a result of Monday's "shambolic" proceedings, Sinn Féin has said.