McClaren is on the brink at St James's Park in the wake of the weekend's home defeat to Bournemouth.
Fans chanted for McClaren, the club's head coach, to be dismissed during the 3-1 loss.
But Bowyer – who left Newcastle a decade ago after a three-year spell on Tyneside – feels the 54-year-old can lift the club out of the Premier League's relegation zone between now and the end of the campaign.
Asked on talkSPORT if he should be sacked, Bowyer said: "No. I think they should just hang on in there with Steve.
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"After Christmas, they brought in a couple of new additions, and I think they were good additions. They turned the corner at one stage. They put some good results together.
"OK, it wasn't a good result at the weekend, and they're still in the bottom three, but I do think they did turn the corner.
"The games I saw on TV, (and) they were at least working hard. Before Christmas, I don't think they were.
"I think (Jonjo) Shelvey's a good buy. They've got someone who can put their foot in and get the crowd behind them."