Shares in AIB shot up to €19.30 in early morning trading but slipped back to €19.23 by mid-afternoon, up 23 cent on the day.
Bank of Ireland rose 4c to €14.94 on the back of AIB's gains, while Irish Life's price slipped 15c to €19.90.
At current prices AIB is worth €16.8 billion, but if Santander wants to buy it they will have to pay up to €25.50 per share, according to Eamonn Hughes, analyst with Goodbody Stockbrokers, which is owned by AIB Group.
That values the bank at over €22bn.
The Financial Times said Santander was ready to return to the acquisition trail following its takeover of Abbey National in Britain two years ago.
In its report, the paper said "people familiar with the matter" have claimed that chairman Emilio Botin was in the market to bolster the Abbey deal.
Santander is understood to have looked at Alliance & Leicester, but opted not to make a bid.
NCB Stockbrokers and Goodbody Stockbrokers see merit in the move from Santander's perspective.
NCB's David Odlum says on the basis of future earnings the shares are under-valued by about 20% relative to the market and could hit €20.50 in the coming months, valuing the bank at €20bn, less than the Goodbody figure.
NCB said its valuation was well underpinned while "bid speculation provides additional upside", he said.
AIB would be an interesting target for Santander for a number of reasons, given its leading position in the fast growing Irish market.
Its British operations which account for 19% of profits could be incorporated into Abbey National, giving both savings and synergies in the British market, NCB said.
If Santander wants to focus on Britain and Ireland it could sell its 23% stake in M&T in the US for €2.5bn.
And its 70% stake in BZ WBK in Poland could be worth over €2bn.
Selling both would make a considerable dent in the takeover cost, said NCB.
If Santander has designs on AIB it would be for its Irish business operations, according to Kevin McConnell of Bloxham Stockbrokers.
While the link-up with Abbey has its merits, the move, if it comes, would be motivated by the bank wanting a presence in the very strong Irish economy, he said.
Right now he said the Santander story is just a matter of "speculation".
Eamonn Hughes of Goodbody Stockbrokers said rumours of a bid did not surprise him, given he had written about a possible takeover at the time the bank announced its results.
He reckons the Spanish bank will have to pay out €25.50 per share, or a 35% premium on the current share price.
He expects the AIB bid rumours will also result in Bank of Ireland being linked to takeover speculation in the near future, he said.