The 26-year-old centre-back has fallen out of favour under Everton boss Sean Dyche and has featured for just 91 minutes in the league so far.
Leeds United, who are eyeing promotion this season, are considering beefing up their defence before the window closes.
Godfrey, who has played under Daniel Farke during his time at Norwich City, is considered a good fit for the squad.
However, Leeds are not the only admirers of the Everton star in the transfer market, as Premier League outfit Sheffield United are among his suitors.
Everton contract and it remains to be seen whether Sheffield United will step up their interest before Leeds seal a deal for the centre-back.