Celtic have been credited with interest in the West Bromwich Albion defender throughout the summer.
The Bhoys have been credited with interest in Dawson throughout the summer, but were behind Premier League sides Burnley and West Ham United in the betting for his signature last week.
It is Celtic (5/1) who lead the way now, however, with Burnley and West Ham drifting to 10/1 and 5/2 respectively.
Dawson was omitted from West Brom's squad to face Bolton on Saturday, and is regarded as being as likely to leave the Hawthorns this summer (5/6) as he is to stay put.
But with both Burnley and West Ham having strengthened in the 28-year-old's position, the chances of either club signing him appear slim.
And that could leave open the door to Brendan Rodgers' Celtic, who unlike their English rivals can sign contracted players until the very end of August.
West Brom's alleged £20 million asking price [Express & Star] may be an issue, however.
Could the West Brom defender be Celtic bound?