Arsenal are said to be preparing a fourth offer for Lemar after their first three were turned down.
Martinez is the third choice keeper at Arsenal behind Petr Cech and David Ospina.
Wilshere returned to training last week after recovering from a broken fibula.
Gunners have been chasing Lemar and are believed to have seen a handful of bids rejected.
Gunners paid €21m to sign the Spaniard from Deportivo La Coruna last summer.
Arsene Wenger addressed the media following the 2-1 defeat to Sevilla in the Emirates Cup.
Frenchman reiterates the former Barcelona forward will stay at the Emirates Stadium.
Nelson and Willock impressed during Arsenal's pre-season tour of Australia and China.
Egypt international has played at centre-back three times during pre-season for the Gunners.
IBTimes UK reported on Tueday that Deportivo had launched an offer worth around to sign Perez.
Both the contract rebel and Shkodran Mustafi due to train on Sunday after post-Confederations Cup break.
The 25-year-old midfielder has a release clause of €40m written into his contract.
A Patrick Vieira-type midfielder and a centre-back are needed says ex-Gunners 'keeper.
Wilshere suffered a leg fracture at the end of his loan spell with Bournemouth last season.
Lemar is set to be Arsenal's third signing of the summer transfer window.
IBTimes UK have already revealed Deportivo have tabled a £9m bid to re-sign him.
The striker was linked with a return to France during talks over Alexandre Lacazette.
Spanish striker wants to leave the Emirates Stadium after feeling unfairly treated by the Gunners.
Sanchez, 28, is keen to reunite with City boss Pep Guardiola.
Wilshere is in the final year of his contract with no talks of an extension yet.
The France international has been linked with moves to Everton and Marseille.
The Arsenal boss is adamant the Chile international won't be sold this summer.