Morison who bagged 22 goals during the season in the Blue Square Premiership is also being linked with Championship side Plymouth Argyle.
With the transfer request being handed in it is believed that Crewe look set to offer £150k for the former Northampton Town striker as early as next week as Steve Holland looks to rebuild the team for a promotion push next season.
Morison told the Hertfordshire Mercury: 'It's nothing against the club. I've been here two years and think I've done well, but I want to play in the league and prove I can score goals there.
'I've always given everything and tried to score as many goals as I can and if I'm still at Stevenage next season I'll do the same.'