The newspaper report that Wilson jumped at the chance to move to the MLS, with the 26-year-old out of contract in the summer and yet to sign a new deal at Ibrox.
The Sun claim that Wilson is now set for a medical with the Rapids, who are part-owned by Arsenal shareholder Stan Kroenke.
It is said the side were keen to boost their defensive options ahead of the new season starting in March.
Wilson has played in every game under Graeme Murty at Ibrox, becoming a key player for the side, but was omitted from the Gers starting XI as they beat Aberdeen 2-0 on Wednesday.
Good move for Rangers?
Wilson looked out in the cold under former boss Pedro Caixinha, but he has come into his own under Murty and has been a real stalwart of the team in recent weeks.
It is not known how far along contract talks were between Wilson and the club, but Rangers have made a mistake not wrapping up Wilson before the Rapids came calling.
To lose such an integral part of your side in January is a big blow, but one that will be somewhat lessened by the arrival of Russell Martin from Norwich.
This is Futbol rating: 3/10
Likely to happen?
If The Sun are to be believed, then this one should be wrapped up soon.
It is no wonder that Wilson wanted to leave after the club failed to offer him an acceptable new contract, with the Rapids most likely tabling a lucrative deal.
If Wilson is undergoing his medical today, expect this one to be completed at some point over the weekend.
This is Futbol rating: 8/10