Gov. Sam Brownback of Kansas vetoed a bill on Thursday that would have expanded Medicaid in his state, setting up a potential showdown next week with a Legislature that, while heavily Republican, has come to favor extending the largely free health coverage to as many as 180,000 additional poor adults.

Although the bill was easily approved in both chambers of the Legislature, supporters would need to muster three additional votes in the House and two in the Senate to override the veto by Mr. Brownback, a conservative Republican.

After the House briefly debated an override on Thursday, the measure was put aside, probably until at least next week. Supporters were hoping to mobilize hundreds of residents who would benefit from Medicaid expansion, as well as hospital executives, clergy members and others who back the measure, to lobby specific lawmakers over the weekend.

“Our plan is to make sure their phones are ringing off the hook,” said David Jordan, executive director of Alliance for a Healthy Kansas, an advocacy group.