The Daily Mail says United goalkeeping coach Chris Woods was in the stands for Stoke's win over Arsenal 10 days ago where 26-year-old Begovic kept a clean sheet.
The Bosnia No 1 was the subject of strong interest from Sir Alex Ferguson a year ago but the upturn in David de Gea's form meant a £12million bid was put on ice.
United are not unhappy with De Gea but want better quality and improved competition in every position for next season. De Gea is not being pushed enough by Anders Lindegaard and United will weigh up options ahead of the summer.
Stoke would want around £15m if they sold but have England keeper Jack Butland to call on should Begovic go.
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