The 23-year-old enforcer has impressed at both club an international level over the course of the past few years, enjoying his breakthrough with Swiss side FC Basel.
United are understood to be searching for defensive reinforcements, with Louis van Gaal keen to bolster his backline after dedicating much of his £150m summer spend to attacking talents.
The likes of Mats Hummels and Ron Vlaar have been linked with moves to Old Trafford, but with the former now a summer target and the latter injured, Dragovic is on the agenda.
Reports claim that the Red Devils have been watching the youngster closely and have been impressed by his showings both in Ukrainian football and the Europa League.
West Ham and Arsenal have also been linked with him, but United are keen to get ahead of their domestic rivals and could do so with an offer before the looming transfer deadline.
Dynamo are reluctant to lose the man they signed in 2013, yet they are ready to offers.