Barkley sustained the injury in their 4-0 FA Cup win over QPR on January 4 and is almost certain to miss the Merseyside derby at Liverpool at the end of the month.
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"It looks like he's going to be out for a while," Martinez told reporters.
"It could be [broken] and that's what we're worried about. It's a very similar incident to the one Leighton Baines had.
"Some can go a couple of weeks and others a bit longer like Leighton. Because it was a bad one it took six weeks."
Baines missed six games after picking up the same injury in the first Merseyside derby of the season at Goodison Park in November.