Maxwell had featured in Rovers’ previous two pre-season friendlies against Spennymoor and Huddersfield but was absent from their latest matchday squad as they held the Millers to a 2-2 draw.
Tommy Rowe filled in at left-back in his absence.
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Doncaster boss Gary McSheffrey said conversations about a deal for the former Rangers youngster, who has represented Scotland at under-19 level, are ‘ongoing’.
He told The Free Press: “He did well considering he’s not had a real pre-season and has been training on his own.
”To come in and play two 70 minutes how he did was good. He’s got good pedigree, he’s got a lot of experience in men’s football already with two loans in Scotland.
"He ticks boxes, but discussions are ongoing.”
Maxwell, aged 20, has already played 75 games in professional football thanks to two loan spells in the Scottish Championship with Queen of the South and Ayr United.
He turned down the offer of a new contract at Ibrox, meaning Rangers would be due compensation if he was to move south of the border.
Left-back remains a priority area for McSheffrey to address this summer after he released Cameron John and Branden Horton at the end of last season.
Meanwhile, former Brighton & Hove Albion youngster Ayo Tanimowo will not be signing for Doncaster Rovers.
The right-back, who played 90 minutes in Saturday’s win over the Terriers, featured for Cheltenham Town in their friendly against Birmingham City tonight.
McSheffrey said: “I had agreed he would play tonight as well but he’s got more of a concrete offer elsewhere.
"Kyle Knoyle being is close to fitness, he’s training on Thursday. Ayo would be a good addition to us but there are more priority positions that need dealing with.
"You can’t keep bringing people in because then you have to get people out.”
McSheffrey wants to begin finalising his squad this week and resolving the futures of the club’s current trialists is another priority.
Jack Dergruchy, Kyle Hurst and Lee Tomlin featured against Rotherham and McSheffrey confirmed an offer is on the table for the latter.
Talks are also ongoing with Hurst, who is still contracted to Birmingham City.