Rovers are eyeing up a loan deal for the Republic of Ireland midfielder.
The midfielder is down the pecking order at Goodison Park, with several players ahead of him in Roberto Martinez’s plans for central midfield.
The former Manchester United man has made just six appearances all season, with four of those coming in the Premier League.
He would have fancied his chances of securing more game time following the injury to James McCarthy, but both Tom Cleverley and Muhamed Besic have been preferred ahead of him, and impressed thus far.
He is likely to fancy a move away from the Blues then to increase his chances of first-team football, and Martinez is reportedly open to a temporary switch.
Securing Gibson would likely provide a big boost to Rovers’ efforts to pull away from the relegation zone this season, with them currently 18th in the table and just five points away from Bristol City in 22nd.