The two sides are expected to resume talks next week.
Soria is attracting strong interest on a free-agent market deep in relievers but thin in proven closers. His 202 career saves rank second among current free agents to another former Tiger, Fernando Rodney. Soria, however, is younger; he won't turn 32 until May.
Detroit has also shown interest in free agent Darren O'Day, a shutdown setup man who saved 12 games over the last three seasons in Baltimore, and is believed to have checked in with the Reds regarding Aroldis Chapman, who's expected to draw a higher price in prospects than the Tigers would want to pay for a reliever one year away from free agency.