The deaths were a "scandal", according to the spokesman who added victims could decide the type of inquiry that is held.
The huge failure, which killed 2,400 NHS patients in the 1970s and 1980s, saw swathes of patients given blood products infected with diseases such as hepatitis C and HIV.
There have been two previous inquiries, but members of Parliament said Tuesday that a new inquest was needed because the previous ones had not gone far enough nor delivered justice to the victims.
"Jeremy Hunt said that 2,400 people had died and it was necessary to establish the causes of this appalling injustice", May's spokesperson said at a briefing, as quoted by the Evening Standard newspaper.
The Government will set up an inquiry into the NHS contaminated blood scandal, Downing Street has announced.
Pressure for an inquiry had grown amid campaigning by Johnson and Andy Burnham, the former Labour MP who is now mayor of Greater Manchester. They have been let down by all political parties and public bodies.
"This inquiry is long overdue and any attempt to lie betray or fob off haemophiliac victims and their families with scraps from the table will be met with the derision it deserves".
A recent parliamentary report found around 7,500 patients were infected by imported blood products. Some of these came from high-risk sources, including prisoners and drug addicts. During that period many of these were imported from the USA where donors were paid, a practice that increased the risk of unsuitable blood.
Julie Morgan, who chairs the Assembly's cross-party group on haemophilia and contaminated blood, said 70 Welsh people died and many were "still suffering".
The Labour leader, Jeremy Corbyn, said there was a need for a "broad, public, inquisitive inquiry".
He said: "Just as with Hillsborough, there must be a "families first" approach at all times".
Hunt told cabinet the inquiry would be United Kingdom -wide and a consultation would take place with those affected to decided what form it should take.
Liz Carroll, chief executive of The Haemophilia Society, said: "The government has for decades denied negligence and refused to provide compensation to those affected, this inquiry will finally be able to properly consider evidence of wrongdoing".