The group wants to hold the events on 4 March in Locharbriggs - where it is based - and nearby Dumfries.
Dumfries and Galloway Council's licensing panel is being asked to issue permission for both events.
However, a parish priest in Dumfries has objected to the march saying it has "no reason whatsoever, historical or otherwise" to be held in the town.
The council has to be consulted by anyone wishing to hold any public procession.
The Locharbriggs Loyal Orange Lodge wants to stage one march through Dumfries at 15:30 on Saturday 4 March lasting for about 45 minutes.
It would follow a route along New Abbey Road, Laurieknowe, Galloway Street, Buccleuch Street, High Street, Shakespeare Street and Nith Place before ending at the Dock Park.
The second procession - starting at 18:00 - would take place for about half an hour through Locharbriggs.
It has been estimated that 75 people would take part in the Dumfries procession and 50 people in the Locharbriggs one.
Fr Jim Hayes of St Teresa's RC church has made a representation against the marches.
He said he was "dismayed" to see a procession being brought into the "heart of Dumfries".
He said he hoped marches of this kind by "any group of any persuasion" would "disappear from Scottish life altogether".
An Orange Lodge in southern Scotland is seeking approval for two processions to mark its 25th anniversary.