Weston is well-known to Dennis Wise and Gus Poyet having worked with them both at Swindon and with Wise at Millwall.
He was taken from the New Den to the County Ground by Wise last summer.
The 20-year-old midfielder had a year remaining on his contract with Swindon, but Paul Sturrock released him last week and Weston looks likely to renew his relationship with Wise in the summer.