Arsenal's Ben White is also on Everton's shortlist for the position, along with Strasbourg's Guela Doue, though the Frenchman is regarded as likely to be too expensive for the Toffees to pursue seriously.
Spence has struggled for regular football at Tottenham despite representing England at the World Cup, with the north London club having strengthened their defensive options significantly this summer through the signings of Andy Robertson and Marcos Senesi, while Pedro Porro has signed a new contract and Destiny Udogie remains first choice on the opposite flank.
That competition has left Spence weighing up his options for next season, with regular Premier League football his priority, reports the Athletic.
Everton, who finished 13th last season, have already been busy strengthening elsewhere this summer, bringing in Hayden Hackney from Middlesbrough and turning Merlin Rohl's loan move into a permanent deal. Identifying a long-term solution at right-back remains a key objective for Moyes as the club look to build a more settled squad ahead of the new campaign.
Jake O'Brien, a natural centre-back, has often been used out of position there following injury problems for both Nathan Patterson and Seamus Coleman, whose Everton contract has now expired, with Patterson also expected to leave this summer.
Spence, tied to a contract at Tottenham until 2029, has featured for England throughout their World Cup campaign and impressed with his pace and versatility, having played at both right-back and left-back for Spurs over the past two seasons.
His World Cup exposure has only heightened interest in his services, with clubs from beyond the Premier League also said to be monitoring the situation, though nothing is advanced at this stage.
No formal approach has yet been made for any of the players on Everton's list, and Tottenham's willingness to sell Spence remains to be tested, but the growing competition for places at Spurs makes a departure from north London increasingly plausible before the transfer window closes.