Mr Varadkar has now decided the money is not there to fund either of the projects, with the announcement due next month.
It is understood they will be deferred, rather than cancelled completely, with the hope that one or both of them could go ahead when the economy improves.
The move will be a blow to the capital, which has an unconnected transport system and no airport rail link. Estimates of the combined cost of the projects have varied but it is believed the total bill would have come to at least €6bn.
A spokesman for Iarnrod Eireann, which is overseeing the Dart project, said he would not comment on the speculation, adding that the organisation was awaiting Mr Varadkar's decision next month.
"It was long flagged [that he would make an announcement]. He said it pretty soon after coming into office," he told the Herald.
The Railway Procurement Agency (RPA), which is in charge of Metro North, said it was also awaiting Mr Varadkar's decision and would not comment on an unofficial report.
However, it is understood the RPA's view is that, realistically external funding could not be raised at the moment.
The two bidders who are vying to build the scheme are believed to be still very much interested in the contract, despite the lengthy delays and threat of deferral.
It is hoped there would be a lot of international interest in financing Metro North -- which is to link the city centre to Dublin Airport -- as soon as the market improves.
Speculation that the projects were in danger has grown in recent months, with the Herald reporting in May the Government was on the verge of ditching Metro North.
Mr Varadkar had said just one of three major capital projects would go ahead.
These were the Dart Underground, Metro North and the link-up of the two Luas lines, called BXD.
The BXD line is still very much a possibility, however.
Meanwhile, a Dublin city regeneration project which became a victim of the property crash has received a major boost. The upgrade for dilapidated O'Devaney Gardens in Dublin 7 has been given planning permission by An Bord Pleanala.
The board granted approval for the first phase of the project and, more importantly, approved the overall plan for the site.
Labour's Emer Costello, a member of the regeneration board, told the Herald she was hopeful building would start within a relatively short period.
O'Devaney Gardens was built in 1954 but is now in a rundown state, with most of the blocks either demolished or de-tenanted.