White cleared waivers Thursday and was sent outright to Double-A Somerset.
White was designated for assignment by the Yankees on Monday.
The Yankees claimed White off waivers from the Rays on Thursday, Gary Phillips of the New York Daily News reports.
The Rays optioned White to Triple-A Durham, Marc Topkin of the Tampa Bay Times reports.
White (elbow) was reinstated from Triple-A Durham's 7-day injured list Aug. 6 and assigned to High-A Bowling Green, with whom he's produced a 3.60 ERA, 1.20 WHIP and 4:2 K:BB across five innings over five appearances.
White (elbow) resumed his rehab assignment in the rookie-level Arizona Complex League on Friday and recorded two outs via strikeout while allowing no runs on one hit and one walk.
White (elbow) has returned to the mound and is making appearances in the Florida Complex League.
White (elbow) was optioned Monday to minor-league camp by the Rays, Marc Topkin of the Tampa Bay Times reports.
White (elbow) threw his fourth bullpen session Wednesday since undergoing Tommy John surgery, RaysRadio.mlb.com reports.