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Johnson, who made his name with the Latvian national side, is believed to have decided to stay with The Glovers after their League Two Championship win, leaving the Millwall chairman to look elsewhere.
Allen, who was an early favourite for the position, has impressed many after taking The Bees to fourth in League One, although they crashed out of the play-offs to Sheffield Wednesday.
Burninge is hoping to make an announcement later this week, having interviewed a short-list of three over the weekend.