The Sunday Mirror says Toffees boss Ronald Koeman and chief scout Steve Walsh have identified Trippier as a replacement for Seamus Coleman, who faces a long absence from football after breaking his leg in two places on international duty.
Trippier, 26, joined Spurs from Burnley for a reported transfer fee of £3.5m ($4.3m), but has been second choice to Kyle Walker at right-back at White Hart Lane.
He has only made seven appearances for the north London club in the Premier League this season.
Everton believe the offer of regular first-team football could persuade Trippier, who has more than three years left on his Spurs contract, to make the switch to Goodison Park.
Spurs are likely to demand a transfer fee of around £6m for the defender's signature, according to the Mirror.
Trippier started and played the full 90 minutes of Tottenham's 4-0 win over Watford at White Hart Lane on 8 April, and was named man-of-the-match for his efforts.
He insisted after the match that Spurs still retained hopes of catching league leaders Chelsea, who hold a seven-point lead at the top of the table.
"We just have to keep performing like today," he was quoted as saying by ESPN.
"We want to take it game by game, we can't affect Chelsea but we have to focus ourselves, take it game by game and see where it takes us."
Goals from Dele Alli and Eric Dier and a double from Heung-Min Son gave Spurs a comfortable win against Watford, prompting manager Mauricio Pochettino to lavish praise on his team's "winning mentality".
"The message for us is to keep pushing and win games, that is our job," he was reported as saying.
"Then what happens this afternoon with Chelsea is their problem. We'll see. If they fail, our job is done because we are there. But it is still up to them to win the Premier League."