Mr Blair would be the first prime minister to give evidence to a Commons select committee if he complies with the inquiry into a raft of appointments that remain closed to public scrutiny.
The Public Administration Committee wants to expose Mr Blair's personal power to select Britain's European commissioners, royal-commission members, archbishops and museum board members. His office also has powers to appoint Regius professors at Oxford, Cambridge and London universities.
The committee decided in private this week to write to the Prime Minister to take part in what it claims will be the biggest-ever investigation into the workings of what remains of the British establishment. It has already embarked on an inquiry into appointments to quangoes but has now decided to widen its remit to include the highest posts of all.
Downing Street officials involved in the confidential appointments process will be asked for evidence. As the Prime Minister has to make his recommendations to the Queen for some posts, it is also possible some Buckingham Palace officials will be called.
Labour, Tory and Liberal Democrat MPs believe that in an age when most aspects of public life are open to vigorous investigation it is an anachronism to retain sweeping prime ministerial patronage.
The Parliamentary Commissioner for Public Standards, the Committee on Standards in Public Life and the Commissioner of Parliamentary Appointments have spent recent years ensuring high standards, but none has a direct role over Mr Blair's appointments.
The Committee on Standards in Public Life, which was first chaired by Lord Nolan, was brought in by the Major government in an attempt to remove the suspicion of sleaze that hung over MPs.
Rhodri Morgan, MP for Cardiff West and chairman of the Public Administration Committee, said it was time that the higher reaches of the establishment submitted itself to the "Nolanisation" that had affected other areas of public life in the past four years.
"We want to shed some light into this murky corner of the democratic system that seems to remain for ever free of the principles of fair play, external assessment and natural justice that apply to other appointments," he said. "These are the last hold-outs of tradition, the ultimate bastion of the great and the good. They sustain this old-boys network that keeps out women and ethnic minorities from posts thought suitable only for white, Anglo-Saxon males.
"The days of sending out the cabinet secretary down to the Reform Club to find someone who fits the bill should be over. These jobs should be advertised openly and monitored to ensure equal opportunities."
The MPs believe that an appetite for reform already exists, as Mr Blair has announced he will give up his sole right of patronage to recommend life peers to the Queen.