The G7 summit, which the US hosts this year, convenes the leaders of Canada, France, Germany, Italy, Japan and the UK to discuss matters of co-operation.
But Mr Trump's invitation to Mr Putin has drawn the ire of the UK and Canada, whose leaders said on Sunday they would not support Russia's readmission to the group.
Russia was expelled from the group - previously known as the G8 - in 2014 in response to its annexation of Crimea.
"Russia was excluded from the G7 after it invaded Crimea a number of years ago, and its continued disrespect and flaunting of international rules and norms is why it remains outside of the G7, and it will continue to remain out," Canada's Prime Minister Justin Trudeau said at a news conference.
Earlier, a spokesman for the UK Prime Minister Boris Johnson said he would veto any proposal to allow Russia to rejoin the group.
Unless Russia ceases its "aggressive and destabilising activity", the UK would not support the country's readmission to the group, Mr Johnson's spokesman said.
"Russia was removed from the G7 group of nations following its  annexation of Crimea and we are yet to see evidence of changed behaviour which would justify its readmittance," the spokesman told reporters.
Neither the UK or Canadian position rules out Mr Putin's attendance of the summit in the US.
Though representatives of non-G7 members have attended the group's summits before, Mr Putin's presence could prove contentious.
In recent years, the UK has had an acrimonious relationship with Russia, which it blamed for a nerve agent attack on a former Russian spy in Salisbury, England in 2018.