The Premier League side, who suffered a 1-0 defeat to the hands of Manchester United at Old Trafford last Sunday, are interested in recruiting the Bristol City centre-half once the current 2015-16 campaign has concluded.
The 26-year-old, who began his career at non-league side Alfreton Town, has scored an impressive six goals and provided two assists in 43 appearances in all competitions for the Robins so far this season.
Flint, who arrived at Ashton Gate from League One’s Swindon Town back in June 2013, made his 100th consecutive start for the Championship club against Rotherham United on Tuesday night.
The report suggests that the towering defender is rated at around £2.5m.
Toffees boss Roberto Martinez has identied Flint as the ideal replacement for England international John Stones, who is said to be a £50m target for Chelsea this summer.