The Brighton striker is 34 years old but has scored 11 Premier League goals this season.
The national boss will name his squad next Thursday for the friendlies with Holland and Italy and is seriously thinking about giving Murray a chance because his striker options are so limited.
Murray being given an England call-up has been discussed by football fans for weeks without anyone giving him a proper chance of making a squad.
England travel to Holland on March 23 and entertain Italy four days later.
Southgate has continually insisted he will reward players who are in good form.
He will make a final decision after this weekend’s games with Murray and his Brighton team travelling to Everton.
Incredibly, he is joint eighth in the League scoring charts alongside Chelsea’s Eden Hazard.
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And Murray has scored the same number of Premier League goals as Tammy Abraham, Marcus Rashford and Danny Welbeck combined.
Abraham, with four goals this season, has failed to find the net on loan at Swansea after being handed his England chance against both Germany and Brazil in September.
Arsenal striker Welbeck has scored three times but not found the net in the League since September while United star Rashford is not a guaranteed starter for Jose Mourinho.
An England call-up for former Carlisle and Rochdale striker Murray would be incredible when you consider he only rejoined Brighton after a disastrous spell at Bournemouth.
Murray scored again in Sunday’s 2-1 win against Brighton but the major concern for Southgate is his obvious lack of pace.
Since Arsenal’s Leslie Compton made his England debut aged 38 in 1950, the oldest player to make his bow since was Bolton’s Kevin Davies.
The striker was aged 33 when he faced Montenegro in 2010 after replacing Peter Crouch as a substitute.
Harry Kane and Jamie Vardy are certain to be in Southgate’s World Cup squad – as will Rashford.
But the Three Lions are still struggling for options outside with 35-year-old Jermain Defoe – who has been picked by Southgate – suffering with his form at Bournemouth.
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