Owen was released by United at the end of last season after three years as a bit-part player with Sir Alex Ferguson's side.
English Premier League teams Stoke City and Everton are understood to be leading the race for the 32-year-old, who only made four appearances for United last season, while Australian outfit Melbourne Heart have emerged as late contenders for his signature.
Via his Twitter account, former Liverpool, Newcastle and Real Madrid star Owen said: "Getting closer to finding a new club. Had a good deal of interest but just need to pick right! Can't wait to get going again."
Owen, who netted 40 times in 89 appearances for England, made a total of 31 appearances during an injury-ravaged spell with United, scoring five goals including a memorable derby winner against Manchester City in 2009.