The 25-year-old is available as a free agent following his release from The Den at the end of last season.
He is believed to have a number of options to consider, with clubs in the Championship, League One and overseas expressing an interest in acquiring his services.
It is, however, Orient that head the queue at present.
The third tier side have ambitious new owners and are hoping to bring in a number of quality additions this summer.
Lowry would add useful Football League experience to their ranks, having taken in loan spells at Plymouth Argyle, Leeds United and Sheffield United prior to linking up with Millwall.