It comes as Jeremy Corbyn said he still expected a “great deal” of movement of people from the EU to the UK.
Setting out their post-Brexit immigration policy, Foreign Secretary Dominic Raab said EU and non-EU migrants would be treated the same from January 2021, including a five-year wait to obtain welfare payments and a surcharge to access health services.
Mr Johnson added: “As we come out of the EU we have a new opportunity for fairness and to make sure all those who come here are treated the same. We will make our immigration system equal.”
Meanwhile, Mr Corbyn said immigration was vital for growth and public services.
In a BBC interview, the Labour leader defended the principle of free movement, which lets EU citizens travel, live, study and work in any member country but which is currently set to end at the start of 2021.
Migration from the EU had “enriched” the country, he said, and this had to be a part of the close economic relationship he wanted to build with the continent going forward.
The Conservatives lead Labour by 10-17 percentage points, four polls late on Saturday showed.
Both parties are expected to publish their election manifestos this week, setting out in detail their full set of policies for government.
Mr Corbyn and Mr Johnson are also due to go head-to-head in their first televised debate on Tuesday evening.
In an interview with the Sunday Telegraph, Mr Johnson said all Conservative election candidates had pledged to vote for his Brexit deal in parliament.
The Mail on Sunday reported Mr Johnson plans a Brexit debate in parliament on December 23 if he wins.
“If we get a majority Conservative government we can deliver and there will be no more wrangling or dither or delay,” Mr Johnson told the Sunday Telegraph.
Mr Corbyn told the BBC he would not form a coalition government if his party fails to secure a majority in parliament.
Asked about possible demands for a Scottish independence referendum in return for the support of the Scottish National Party (SNP) if Labour is short of a majority, Mr Corbyn said: “We are not doing deals with anybody.”
“We are not forming coalition governments, we will put forward the programme on which we will have been elected,” he added.
“The SNP will have a choice, do they want to put Boris Johnson back in … or are they going to say a Labour government will deliver for Scotland.”