The 23-year-old will now go on to a week-to-week deal at Glanford Park.
Iron boss Brian Laws had wanted to tie down the full back to a new long-term deal and he was disappointed at the player's decision to turn down the deal.
"Andy has turned down the deal but he has stressed he will still give one hundred percent while he is at the is football club," Laws told the club's official website.
The news could alert a number of clubs to Dawson's availability.
