WASHINGTON -LRB- CNN -RRB- -- Republicans were facing pressure Tuesday to vote for a rollback of across-the-board cuts in Medicare payments to health providers after a major doctors ' group said the cuts could lead to a `` meltdown '' of the government 's health care system for the elderly . Doctors say they would take on fewer Medicare patients if the cuts go into effect . On Monday , Senate Majority Leader Harry Reid , D-Nevada , called several Republicans trying to persuade them to support a bill that would reverse a 10.6 percent cut in Medicare payments to doctors . The cuts in Medicare payments -- part of a scheduled cost-saving formula -- went into effect July 1 , although the Bush administration said it will hold off on processing claims until mid-July to give Congress time to reach a compromise . The Medicare system pays for the health care of roughly 40 million elderly Americans . Rising health care costs have made Medicare a growing part of the federal budget , and the stress on the system is increasing as more baby boomers reach retirement age . Similar cuts have been scheduled to go into effect on July 1 in previous years , but Congress always has voted to stop the cuts before they went into effect . Gerald Harmon , a family physician who practices in Pawleys Island , South Carolina , said the cuts could lead to doctors taking fewer Medicare patients , making it difficult for the program 's elderly patients to get the care they need . `` This Medicare access problem is a real issue , not just a political football , '' said Harmon , who said 35 percent of his patients were eligible for Medicare . `` It affects your dad , when he 's sick . It affects my patients in my practice . This has to be addressed . '' A vote on the bill last week fell one shy of the 60 needed to clear a Republican filibuster and advance in the Senate . Senate Republicans joined the White House in objecting to the Democratic-back bill because it paid for the increased payments to doctors by trimming government support for private insurance programs that provide coverage to Medicare patients . The private Medicare programs are a top policy initiative for Republicans . The issue has become the subject of an intense election-year lobbying and advertising campaign aimed primarily at its GOP opponents . The American Medical Association , a powerful doctor 's group , said its members may be forced to reduce service to Medicare patients if the cuts go into effect . According to a recent association survey , 60 percent of physicians will be forced to limit the number of new Medicare patients they can take on if the July 1 cuts go into effect . `` We stand at the brink of a Medicare meltdown . ... For doctors , this is not a partisan issue -- it 's a patient access issue , '' AMA President Nancy Nielsen said in a statement after the Senate vote last week . The AMA ran radio and TV ads over the July Fourth congressional recess targeting 10 Republican senators , seven of whom are up for re-election . The AARP , the nation 's largest organization of retired persons , and other groups also are weighing in against the cuts . Reid has `` no idea '' if the pressure has changed any minds , but the `` proof will be in the pudding '' when senators vote again to break the filibuster , said Jim Manley , a spokesman for the speaker . That vote is likely to occur Wednesday . `` There is clearly a lot of pressure , '' Sen. Max Baucus , D-Montana , chairman of the Senate Finance Committee , said Monday . Baucus said he has talked to several Republicans and `` there are a few '' who could switch their votes . But GOP lawmakers and aides gave no indication the intense lobbying had changed any minds , and even if the bill does pass the Senate , it may still face a presidential veto . Republicans `` are holding out for a compromise that can pass and the president can sign , '' said a top GOP leadership aide . At the heart of the dispute is the decision by Democratic lawmakers to pay for the `` doctors ' fix '' by cutting funds for Medicare Advantage , a program administered by private insurance companies that is generally favored by Republicans and opposed by Democrats . The program , which has more benefits than traditional Medicare , is also more expensive . Republicans -- joined by the insurance industry -- fear Democrats are trying to weaken the Medicare alternative by draining funds away from it . President Bush has threatened to veto the legislation because of the cuts to Medicare Advantage -- so that even if Democrats get one additional vote to beat a filibuster , they will need several more to override a veto . The House of Representatives passed the bill 359-55 with broad Republican support in June , but the two-thirds vote needed to override a presidential veto appears , at the moment , to be an insurmountable barrier in the Senate . `` Our goal is to get a result , '' said Senate Minority Leader Mitch McConnell , R-Kentucky . `` A vetoed bill is not a result . '' McConnell on Monday pressed for a monthlong extension of the current fee schedule so senators could work on a compromise . CNN 's Ted Barrett and Miriam Falco contributed to this report