The Toffees are said to have been close to a deal for Martina in the summer, as manager Ronald Koeman - who signed the Southampton right-back while in charge at St. Mary's - sought to add competition for Seamus Coleman.
Southampton had previously added Jeremy Pied to their right-side options, which looked to have paved the way for Martina to move, however the Frenchman was sidelined with a serious injury not long after.
Everton, who were without Coleman at the time, instead turned to Mason Holgate in the Irishman's absence, and the youngster's subsequent displays have reportedly convinced Koeman to shelve his plans to add another body in that department - if only for the coming transfer window.
Another right-back, Jonjoe Kenny was praised by manager David Unsworth for his performance in Everton Under-23s' draw against the Saints on Monday, and proved his credentials when called upon at the end of last season.
While Callum Connolly is another who Koeman could turn to if required, although Tyias Browning is said to be some way from a return to fitness.