WASHINGTON—Some big-money Republican donors, frustrated by their party's handling of the standoff over the debt ceiling and government shutdown, are stepping up their warnings to GOP leaders that they risk long-term damage to the party if they fail to pass immigration legislation.

Some donors say they are withholding political contributions from members of Congress who don't support action on immigration, and many are calling top House leaders. Their hope is that the party can gain ground with Hispanic voters, make needed changes...