The 32-year-old, who won gold for England in the 5,000 metres at the Manchester Games four years ago, has not fully recovered after bruising her right foot in January.
"Paula hit a rock when out training in January and developed a bruised foot and it is taking a long time to settle down," Lough said in a statement.
"In particular it has been causing problems on the track, constantly flaring up and she cannot do proper flat out training sessions.
"It has reared up again and we feel it would be too much of an unnecessary risk, to compete in Melbourne. The problem is not affecting her marathon preparations."
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Radcliffe will attempt a fourth victory in the London marathon on April 23.