Gallagher, 26, joined Tottenham in January in a 35 million pound deal from Atletico Madrid but is unlikely to remain at the club should they drop into the Championship. Spurs sit 17th in the table heading into the final stretch of the season, with only two points separating them from West Ham United in the relegation zone.
Moyes is determined to strengthen his midfield despite Everton's own encouraging campaign, with the club fighting for a European place. Gallagher's Premier League experience and work rate fit the profile the manager is targeting, says The Sun.
The Goodison Park club are also monitoring McGinn, whom they enquired about last summer without making a formal offer. Villa could be open to selling if they are successful in recruiting Fulham's Harry Wilson, who is available as a free agent at the end of the season.
Everton will additionally approach Chelsea about taking Delap on loan with an option to buy. The striker's future at Stamford Bridge is uncertain following the return of Jackson and the imminent appointment of Xabi Alonso, who will oversee a summer of significant squad reshaping.