The 29-year-old keeper who arrived at Crewe last week has impressed in training and also played in a reserve game against Birmingham even though Crewe lost 5-3.
If he accepts Crewe's offer he will be contracted to the club up until January 2010 when the transfer window repoens.
The Calais born keeper started in the English game with Barnet after failing to make an appearance for Racing Club De Lens during three years at the club. Since then Bossu has been at nine clubs with Aberdeen being the last where he made three appearances for the Dons.