WASHINGTON—A growing number of congressional Republicans want to use a contentious budget procedure to dismantle the Affordable Care Act, according to senior lawmakers and aides involved in discussions over the strategy.

A final decision could be weeks if not months away, but Republican lawmakers and aides say there is broad enthusiasm for using the process known as reconciliation to try to repeal the 2010 health-care law and potentially replace it with a GOP alternative. Another contender, looking less likely now, is using...