HAVANA — Big businesses and politicians are putting muscle into their lobbying to end the US embargo of Cuba. A new bipartisan lobbying group called Engage Cuba debuts this month with backing from more than a dozen Fortune 500 companies, said James Williams, who heads the effort. And a political action committee called New Cuba PAC is raising money. Engage Cuba, whose staff includes veteran Democratic and Republican operatives, will focus on striking down the ban on American tourism to Cuba. President Obama has loosened the requirements for official permits for Americans to fly to Cuba to engage in 12 permitted activities, ranging from educational travel to musical performances. A growing number of pro-business Republicans are siding with anti-embargo activists. ‘‘We’re seeing an increasing number of Republicans seeing that this is a better route and that this is an opportunity for Republicans to put their stamp on this policy’’ said Ricardo Herrero, who works with Williams’s groups and heads his own new anti-embargo group, CubaNow.