Enrique, who has missed the last two matches with a knee problem, is also pushing for a recall, with the defender expected back in training this week.
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Barton has had a frustrating time with injuries this term, having been sidelined with a broken metatarsal picked up at former club Manchester City in January.
Shearer: "Joey did a little bit of training on Monday.
"We hope that he will be fully training by Thursday or Friday. And if everything goes well then we might have him available for Portsmouth.
"Hopefully Jose Enrique will be training Wednesday or Thursday."