Mr Carlaw said he "ready for the fight" after leading the party's general election campaign north of the border.
Frontbencher Michelle Ballantyne, the party's social security spokeswoman at Holyrood, is also widely expected to stand.
It comes after Ms Davidson's sudden resignation from the role last August.
Mr Carlaw already has the backing of Holyrood frontbenchers Liam Kerr and Rachael Hamilton, who are jointly chairing his leadership campaign. Other key party figures who had been linked with a leadership bid, including Murdo Fraser and Adam Tompkins, are also backing Mr Carlaw.
"As nominations open for leader of the Scottish Conservative & Unionist Party this morning, I can confirm I'll be standing," Mr Carlaw tweeted today.
"We must take the fight to Nicola Sturgeon and the SNP, broadening our platform and diversifying our party. I'm ready for the fight."
The hunt for Ms Davidson's successor was put on hold after Prime Minister Boris Johnson called a snap general election.
The Tories lost seven of the 13 seats they had held in Scotland, with the campaign focused heavily on opposition to First Minister Nicola Sturgeon's plans to hold a second independence referendum.
Writing in the Sunday Times newspaper, Mr Carlaw said the next year's Holyrood elections must be "about far more than our constitutional settlement".
He called on Tories to raise their game, saying: "We need to be bold and fearlessly champion solutions to the challenges of the 2020s that Scotland will face - to create opportunity in housing and education, secure a sustainable future for our publicly-funded NHS, deliver - not just promise - on the environment and transform our post-Brexit economy.
"And to succeed, Scottish Conservatives at Holyrood need to look more like the Scotland we seek to represent, and to embrace procedures that deliver this.
"In short, we need to ensure that while we have many more new MSPs joining our team in 2021, all are typical of the new generation of Conservatives representing their communities at all levels, diverse in every sense, talented, experienced, of all ages and backgrounds."
Nominations close at noon on Friday January 17, with those looking to be the next leader needing to secure the support of at least 100 party members.