Lockett agreed Monday with the Reds on a minor-league contract, Bobby Nightengale of The Cincinnati Enquirer reports.
The Blue Jays are expected to release Lockett after he agreed in principle on a contract with the Korea Baseball Organization's Doosan Bears, Jee-ho Yoo of the Yonhap News Agency reports.
Lockett was claimed off waivers by the Blue Jays on Monday.
Lockett was claimed off waivers by the Mariners on Tuesday.
Lockett was designated for assignment by the Mets on Friday, Anthony DiComo of MLB.com reports.
Lockett is expected to start Friday's game against the Phillies in place of Jacob deGrom (finger), Mike Puma of the New York Post reports.
Lockett (back) was activated from the 10-day injured list Tuesday, Tim Britton of The Athletic reports.
Lockett was placed on the 10-day injured list with lower back discomfort Monday, Anthony DiComo of MLB.com reports.
Lockett gave up three hits over three scoreless innings in Monday's Grapefruit League game against the Marlins. He struck out two.
Lockett was recalled from Triple-A Syracuse on Saturday, Tim Healey of Newsday reports.