The 32-year-old German midfielder completed his transfer from United to the MLS on Tuesday, joining the Fire on a one-year deal.
Schweinsteiger had featured just four times for Mourinho this season, indicating that there was perhaps some bad blood between he and Mourinho, but Rodriguez has quickly quashed those rumours.
"He's (Schweinsteiger) has had nothing but good things to say about United," he told Sky Sports.
"Clearly he had a great affection for the fans, but also for the players and management.
"He felt he could have played more as he's 32 and in good shape. In fact, this process to bring him to Chicago could have been done earlier were it not that United wanted him to stay at the club during the early part of the season.
"United were competing on many fronts (earlier in the season) and Bastian was regarded as a valuable asset by United.
"Obviously as the season goes on, rivals have dropped away. Now is a good time, United have fewer competitors and a summer transfer is not good for us. We made it happen."
Rodriguez also stated that United were a pleasure to deal with throughout the negotiating process: "Manchester United were very gracious and (United executive vice-chairman) Ed Woodward was particularly helpful to get the deal done.
"We believe Bastian will make us a better soccer team, he'll be a role model and a leader for us."