“That’s the billion-dollar question,” said Robert Blizzard, a Republican pollster who has studied the pattern of how a president’s support has affected Senate races over the last decade. His findings: Precious few candidates can outrun the president by more than a half-dozen percentage points — a worrisome fact for Democrats when Mr. Biden’s approval has fallen below 40 percent nationally.

“The president’s approval rating acts as a weight on their party’s nominee,” Mr. Blizzard said. “Gravity is going to apply at some point.”

So far, Senate Democrats have been buoyed by a cash edge, some strong candidates and the fact that Republicans have nominated a series of first-time candidates — Herschel Walker in Georgia, Dr. Mehmet Oz in Pennsylvania, Blake Masters in Arizona — who have struggled to find their footing, have faced questions about their past and have generally been unable to keep the 2022 campaign focused on unhappiness with Democratic rule in Washington.

Republican strategists involved in Senate races, granted anonymity to speak candidly, say that those three candidates — all of whom were endorsed by Donald J. Trump in the primaries — are falling short of expectations.