Corrective Services Minister Bill Johnston says about 20 MPs from Progressive Labor have agreed to endorse Mr Cook for the state's top job when Mark McGowan steps down later in the week.
"We met with Roger first and he explained his vision for Western Australia and the Labor Party and that's a vision we share," he told reporters.
"It's (also) a vision that builds on the strength Mark McGowan has left us, and it takes advantage of Roger Cook's experience in a united team with Rita Saffioti to take the Labour Party forward."
Mr Johnston said no demands were made of Mr Cook in exchange for the right faction's support.
"We've set no conditions. We're just so pleased he prepared to be engaged," he said.
"He's got the experience. He's got the talent to be a great premier of this state."
Mr Cook, a member of the left faction, is now all but certain to step into the leadership role.
His main rival for the Labor leadership had been Health Minister Amber-Jade Sanderson, also from the left faction.
But she dropped out of the race on Tuesday after MPs tied to the Australian Manufacturing Workers Union swung their support behind Mr Cook with the factionally unaligned transport minister, Ms Saffioti, as his deputy.
Their decision widened a split within the left after Ms Sanderson earlier won the backing of a slim majority of MPs tied to the United Workers Union.
Mr Cook is poised to become the state's 31st premier.
The 57-year-old has served as Mr McGowan's deputy since Labor formed government in 2017.
He was also health minister at the height of the COVID-19 pandemic before assuming his current portfolios of state development, jobs and trade.
It is likely he will have to balance competing demands in forming his first ministry to a greater degree than the factionally unaligned Mr McGowan, whose authority stemmed largely from his extraordinary personal popularity.
Ms Sanderson said she had decided to withdraw "in the interests of unity and stability".
"I am fiercely proud of everything that this Labor government has achieved over the past six years," she said in a statement.
"I respect the process and the views of the entire WA Labor team."
A protracted leadership battle had loomed before Ms Saffioti met with Mr Cook on Tuesday and agreed to serve as his deputy.
"The deputy premier and I have been parliamentary colleagues for 15 years and cabinet colleagues for six years," Ms Saffioti said.
"What we offer above all else is stable leadership, which is what our state needs right now."
The entire parliamentary party is expected to meet early next week.
8°C