According to The Sun, the Light Blues wanted to bring Gilmour back to the club in the winter transfer window as well.
The Scotland international was excellent in his country's 0-0 draw with England in the Euro 2020 group stages this month.
While Rangers want to secure him quickly, they may have to wait until they secure Champions League qualification before any move is finalized.
Gilmour would only go to Rangers if they were in Europe's top competition, as his first preference is to stay in the Premier League.
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