The French-born Mbolhi, 24, named in Algeria's preliminary squad for next month's World Cup in South Africa, has flown to England and will not take part in Slavia's last two matches this season.
"I've talked to (United manager) Sir Alex Ferguson and he's going on trial," club president Ventsislav Stefanov told the Bulgarian state agency BTA.
Mbolhi, who also represented France at youth level, joined Slavia in June and was voted the Bulgarian league's best keeper seven months later.
The Red Devils continue to be linked with a string of stoppers as a future replacement for 40-year-old Dutchman Edwin van der Sar.
Reserve goalkeeper Ben Foster revealed on Wednesday that he is to seek a move away from Old Trafford after growing frustrated at his lack of regular action between the sticks for Ferguson's side.
