"Jermain's recovering particularly well. It seems that he has a good recovery time from this type of injury... so it doesn't look as serious as first thought," Allardyce said at his pre-match news conference on Thursday.
"It looks like, the way he's progressing he will be okay for next weekend (against Watford)."
Midfielder Sebastian Larsson is also unavailable for the game against fourth-placed Arsenal and has flown to Barcelona for treatment on an injury to his medial ligaments, Allardyce said.
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(The story was refiled to correct Defoe's Premier League goals to four)
(Reporting by Simon Jennings in Bengaluru; editing by Justin Palmer)
