The cross-party group said speaking English was "the key to full participation in our society and  economy".
They also said ministers should consider letting different parts of the UK set their own immigration policy.
The government said it was spending Â£20m on English language provision.
Last month a report by Dame Louise Casey warned of "worrying" levels of segregation in some areas and called for more English classes for isolated groups.
The report by the All Party Parliamentary Group on Social Integration urges the government to go further, saying all immigrants should have either learnt English before coming to the UK or be required to sign up to classes when they arrive.
The group said integration should begin upon arrival in the UK and described speaking English as a "prerequisite for meaningful engagement with most British people".
The government has promised new migration controls once the UK leaves the EU but has not yet set out a detailed model.
According to the cross-party group, "substantial" immigration powers should be devolved to the UK's nations and regions, with devolved governments or city regions setting quotas.
4.1m
population increase 2001-11
50%+
of the increase is due to immigration
2.8m people in Britain are Muslim - the biggest religious group after Christians
70-85% of the populations in some wards in Blackburn, Birmingham, Burnley and Bradford are Muslim
511 schools in 43 areas take more than 50% pupils from Pakistani or Bangladeshi backgrounds
27% of births in 2014 were to mothers born outside the UK
It said replacing the current "one-size fits all" approach would lead to a "more positive" public debate about immigration.
Addressing the "economic and cultural needs" of an area would "have a positive knock-on effect on the public debate on immigration", the report said, adding that this "could instil confidence among members of the public that the immigration system works for their area, and give incentives for politicians to actively make the case for immigration in their area".
Pointing to a similar model in use in Canada, where provincial governments can set region-specific requirements for immigrants, it said visas could be issued specifically for certain regions or certain sectors.
The report also called for a new national strategy to integrate immigrants, including "awareness of the host country's laws, traditions and culture" and access to the labour market, and for councils to set up local integration action plans.
Net migration is well above the government's target of below 100,000, and ministers have said they will reduce it after the UK leaves the EU and is no longer subject to freedom of movement rules.
Labour MP Chuka Umunna, who chairs the all-party group, said a "meaningful" integration programme would be needed when the rules are changed after Brexit.
He said: "It's clear that immigration has impacted on different communities in different ways and the pace of change has alarmed many.
"The government has a duty to address the lack of integration of immigrants if it is to address this. Failing to do so has left a vacuum for extremists and peddlers of hate to exploit."
The Home Office said it was not planning to introduce local visa arrangements, but the department said it had made funding available for more English lessons.
A government spokesman said: "Our country has long been home to lots of different cultures and communities, but all of us have to be part of one society - British society."
He said Â£140m was available through a "controlling migration fund" to help councils manage the impact of immigration.
He added: "However, we must also recognise that uncontrolled, mass immigration makes it difficult to maintain social cohesion and puts pressure on public services.
"Our priority is to build an immigration system that works for everyone in the UK and delivers the control we need."
A group of MPs and peers says migrants should be expected to learn English before coming to the UK, or attend language classes when they arrive.