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England have been searching for a new coach since Chris Silverwood left the role in the wake of the Ashes defeat over the summer.
The Daily Telegraph reports that Dunedin-born McCullum "has emerged as a strong contender" for the England job. South Africa's Gary Kirsten and Australian Simon Katich are other considered other contenders.
McCullum currently coaches the Kolkata Knight Riders who sit seventh on the Indian Premier League table, two wins outside the top four.
McCullum has previously been tipped as a potential contender as the England limited overs coach.
His noted aggressive style as Black Caps captain and now coach with the Knight Riders would marry well with new England test captain Ben Stokes who will lead the team against the Black Caps in the three test series starting June 2 at Lord's.
England are currently ranked sixth in the world test rankings and bottom of the World Test Championship standings with one win from their last 12 tests.