Boris Johnson will give the final go-ahead to the first phase of the controversial HS2 high speed rail link early this week – despite fears over spiralling costs and strong opposition from at least 60 Tory MPs.

The prime minister is expected to make an announcement to parliament on Tuesday approving construction of the line between London and Birmingham, two days before conducting a wide-ranging reshuffle of his cabinet.

Downing Street made it clear last night that the green light for HS2 would be accompanied by details of a range of other infrastructure projects, particularly for the north of England.

A senior government source said: “What we want to do is show that infrastructure is about all projects –not just the biggest ones – for all parts of the country.”

It is expected that Johnson will stop short of giving full approval to the second phase of HS2 – lines north of Birmingham to Manchester and Leeds, saying more analysis needs to be done on the costs.

The announcement will follow years of argument over the cost and environmental impact of the line. The estimated costs of HS2 between London and Manchester and Leeds have more than doubled from £56bn in 2015 to as high as £106bn, according the latest Department of Transport projections.

As well as increasing capacity, HS2 would cut journey times between London and Birmingham from about one hour 20 minutes to 45 minutes and between London and Manchester from two hours eight minutes to one hour eight minutes. Between London and Leeds, journey times would be cut from around one hour 58 minutes to 57 minutes.

The latest estimated opening times are 2028-2041 for the first phase from London to Birmingham, and 2035-2040 for completion of the lines to Manchester and Leeds. But the National Audit Office warned recently that even these revised opening dates would not be met unless the government moved fast to approve phase one.

The go-ahead will put Johnson at loggerheads with many of his own MPs, including several who were elected in December in previously Labour-held seats, and who believe the money would be far better spent on a range of other measures to improve connectivity across the entire country.

Johnson met groups of concerned Tory MPs last week to reassure them that HS2 would be accompanied by other infrastructure projects, as well as improvements to bus services and broadband for local areas.

A No 10 source said: “The PM understands the responsibility he has to deliver for everyone who put their trust in the Conservatives in the election. That means transforming the transport and infrastructure links in local areas, particularly in the north. For some this will mean big, ambitious projects – but he has been struck also, by the small changes that will make an even bigger difference to the everyday lives of people across the country – be they roads, rail or other projects.”

Downing Street made it clear that Transport Secretary Grant Shapps would be kept in post in the reshuffle to champion improvements to rail, road and bus services.

Shapps will this week order West Midlands Trains to deliver a £20m package of improvements for passengers, making up for recent poor performance. The investment will go towards improving services and reducing delays, extra compensation for passengers and recruiting extra drivers.