A positive test meant that Zaha, who had appeared in Palace's previous 59 Premier League games, missed Monday night’s trip to Burnley, where the Eagles lost 1-0 thanks to Chris Wood's early goal.
Palace struggled without Zaha's quality and creativity in the final third as the hosts sat deep at Turf Moor.
With Newcastle visiting Selhurst Park on Friday, Hodgson is hopeful that the 28-year-old - who he says is not feeling particularly unwell - will return a negative test this week.
"We've known about it for a few days," said Hodgson. "We didn't want to go out and broadcast it to the world because we rather hoped a second test would free him to play.
"He's had to self-isolate, unfortunately. Now we'll have to wait for that test to hopefully free him because fortunately he's not feeling particularly unwell, it is just that he failed a test."
After summer investment and an impressive start to the season lifted expectations in south London, many were hopeful that Palace could kick on and challenge in the top half of the Premier League.
However, winger Andros Townsend says the Eagles must come away with all three points from games against the likes of Burnley if they are to put any sort of run together, with the Clarets having not won a league game all season before Monday night.
He said: "We created chances but weren't clinical. Nick Pope is an England international, you expect him to make top saves. We were pushing, throwing everything at them creating carnage, but we could not get that all-important goal.
"If we have aspirations of finishing in the top 10, these are the games you have to be winning."