Work on the stadium has been marred by delays and scandals that saw its constructor sacked this summer.
"It's a very sad story," Russian news agencies quoted Putin as telling Infantino at the Kremlin, referring to the scandal-plagued construction of the Saint Petersburg stadium that will host the Confederations Cup in 2017 and a World Cup semi-final in 2018.
"The builders promise to fix everything by the end of the year."
Work on the stadium, which got underway in 2007, has been marred by delays and scandals that saw its constructor sacked this summer.
Problems with a retractable pitch at the futuristic 68,000-seater stadium have rendered it unstable and unsuitable for play.
Authorities have pledged that the problem would not cause any more delays to the stadium's construction, which is set to be completed by the end of the year.
Earlier this month, the former deputy governor of Saint Petersburg, Marat Oganesyan, was arrested over a fraud scheme with a firm that was supposed to provide the stadium with a video scoreboard.
The venue, which will be home to Zenit Saint Petersburg, is one of 12 that will host matches during the World Cup.