The Magpies coach is considered to be on borrowed time.
Frustrations reached boiling point on Saturday when top-flight new boys Bournemouth humbled them with a 3-1 home defeat, in which the Newcastle fans chanted for McClaren's removal.
He signed Jonjo Shelvey, Andros Townsend and Henri Saivet in the January transfer window to strengthen United's chances of survival, and Bowyer - who spent three years at the club until 2006 - says that these players can help them turn a corner shortly.
"No. I think they should just hang on in there with Steve," he said on Talksport, as quoted by the Shields Gazzette.
"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."
It does not get much easier for Newcastle, who visit league leaders Leicester City next Monday, before hosting bitter local and relegation rivals Sunderland six days later, with the Black Cats one point and two places ahead of them in 17th.
Although it does remain to be seen whether McClaren will be in charge for either clash.