The former Crewe star has made an excellent recovery from a hernia operation and will be available for the games with Southend and Darlington.
Unsworth played 60 minutes of a reserve fixture against Shrewsbury without suffering an adverse reaction.
"Lee is a first-class athlete and his operation and rehabilitation went well," said Barrow.
But Canadian mid- fielder Terry Dunfield won't be competing at this summer's Olympic Games in Athens.
Dunfield had to give his country's games at the CONCACAF Olympic qualifying tournament in Mexico a miss to help the Shakers fight for survival.
In his absence the Canadians lost 1-0 to Honduras, 2-0 to the USA and 2-1 to Panama.
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