Mr Agius quit earlier this month due to the Libor fixing scandal, but then had to take on the running the company following the departure of chief executive Bob Diamond.
Barclays has been under pressure to appoint someone from outside the bank.
Several top shareholders had protested at the prospect of Sir Michael’s appointment, according to the Reuters news agency.
Favoured candidates have begun to emerge for both top jobs at Barclays, BBC business editor Robert Peston says, although the selection process is still at an early stage.
The former head of the civil service, Lord O’Donnell, is the board of the bank’s preferred choice as new chairman.
And Bill Winters, the former JP Morgan banker, is the front-runner to succeed Bob Diamond as chief executive of the bank.
It is not clear whether either would actually take it, our correspondent says.
But whoever gets appointed can expect to come under close scrutiny from the regulatory authorities, including the Bank of England, whose governor Sir Mervyn King played a crucial role in the departure of Bob Diamond.
Sir Michael’s decision was announced by Easyjet, for whom he is chairman. He is also chairman of BT Group.
He would have had to give up both of these roles if he had taken the top job at Barclays.
Easyjet said that Sir Michael had informed its board and Mr Agius that he did not wish to be a candidate to fill the vacancy at Barclays.
It may mean that Mr Agius, who has been chairman for five and a half years, has to stay on even longer than originally planned while a successor is sought.
Remaining candidates are said to include the former cabinet secretary Gus O’Donnell, and Glen Moreno, the chairman of publishers Pearson and former deputy chairman of Lloyds Banking Group.
Mr Agius, Mr Diamond and Barclays chief operating officer Jerry del Missier all stepped down at the beginning of July after the bank was fined £290m by the UK and US regulatory authorities for manipulating Libor – a key benchmark interest rate in the global financial markets – over many years.
Reports have said that Barclays is set to appoint financial law expert Anthony Sal this week to head a promised internal inquiry into the Libor scandal.