In late April, when Mike Massaro set out to get $40 million to $75 million in funding for his payments start-up, Flywire, he contacted a small group of investors he already knew. But word quickly got around, and other investors flooded his inbox with $200 million of investment offers, half of which he turned down.

Gusto, a payroll and benefits software company, raised $140 million in July, but could have done five times that, according to Joshua Reeves, its chief executive and founder.

Convene, a real estate services start-up, recently obtained $152 million and turned away more than $100 million of additional investment. Soon after, another wave of hopeful investors called, asking if the company would be looking for more financing, according to Ryan Simonetti, Convene’s chief executive.

Start-ups raising $100 million or more from investors — known as a mega-round in Silicon Valley — used to be a rarity. But now, they are practically routine, producing a frenzy around tech companies with enough scale and momentum to absorb a large check.