Donovan, who has 57 goals in 157 matches with the US side, spent four seasons with the San Jose Earthquakes from 2001-2004 then played for the Los Angeles Galaxy from 2005-2014, coming out of retirement for six regular-season matches and three playoff contests earlier this year.
ESPN, Sports Illustrated and Fox Soccer reported that Donovan had been offered a two-year deal by Real Salt Lake.
Donovan will turn 35 on March 4, the day Real Salt Lake plays host to Toronto FC in its 2017 season opener.
Donovan ended a 21-month retirement last September, in part because injuries had hindered his former club, and added a goal to his MLS record total, now at 145.
In all, Donovan was a six-time MLS Cup champion, taking the title with San Jose in 2001 and 2003 and with the Galaxy in 2005, 2011, 2012 and 2014.
After an ill-fated stint with Bayer Leverkusen in 2000, Donovan found a measure of success in Europe with six matches for Bayern Munich on loan in 2009 and 17 with Everton over two stints in 2010 and 2012.