The Welsh maestro was Brendan Rodgers’ first major signing when he took over the Reds in 2012, and was hailed as the player to implement his slick passing style at Anfield.
Yet Allen has failed to hold down a regular starting berth on Merseyside, despite making a promising start to his career at the club.
This has alerted Martinez, who is known to be a massive fan of the silky playmaker.
The Spaniard gave the pint-sized midfielder his debut when in charge of Swansea, and has since tracked his career.
Despite the bitter rivalry between the two local sides, Everton are confident that they can tempt Allan to make the switch.
Liverpool paid £15m for their man, and with three years remaining on his contract they are likely to demand at least £10m, despite him not being a key member of their first-team.
Everton have not purchased a player from Anfield for 22 years, with the last man making the short trip from the red side of Stanley Park to the blue one being Gary Ablett in a £750,000 deal.