Mr Swinney is the only declared candidate so far to succeed Humza Yousaf, who announced he would stand down after ending a power-sharing agreement with the Scottish Greens.
The abrupt end of the Bute House Agreement came two days after Mr Yousaf said he had no intention of doing so, and left him facing two votes of no confidence.
The move angered the Greens, who had previously criticised the Scottish government for scrapping key climate targets and pausing puberty blockers for under 18s referred by the country’s specialist gender clinic.
Last year, Mr Yousaf only secured a narrow win over Kate Forbes in an often fractious SNP leadership contest – despite being Nicola Sturgeon’s chosen successor.
Mr Swinney – who secured Kate Forbes’ backing and has offered her a “significant” role in his government – said he wants to bring the party together after the last “difficult” leadership election.
He also said he is “physically and mentally ready” for the role, after he decided not to stand in the 2023 leadership contest.
Mr Swinney previously led the party for four years in opposition from 2000 to 2004.
If elected, he said he would seek to govern on a “moderate centre left” platform, and that he would be open to working with any other political party in the parliament to pass legislation.
