Dutch rival ING, Spain's BBVA and France's BNP Paribas have also expressed an interest - directly or through advisers - in either exploring a full merger with ABN AMRO or buying some of its large non-Dutch businesses, the sources said.
Amsterdam-based ABN AMRO owns large retail banks in North America, Brazil and Italy and the investors, which include British hedge fund TCI, have said it should consider a sale or breakup to boost shareholder returns.
Any talks being held by ABN AMRO with other banks are at a very early and uncertain stage, the sources said.
An announcement from Barclays was expected in a statement to the London Stock Exchange, The Financial Times reported.
The Times and The Daily Telegraph British newspapers also expected such news, while The Independent said both ABN AMRO and Barclays were set to confirm their early-stage talks.
The Sunday Times said Barclays was "keen to act as a white knight to save ABN from its current problems".
Another source said ABN has reached no decision on how to proceed and a quick deal before a shareholder meeting being held next month was unlikely unless the bank was made a hugely compelling offer in the meantime by one of its suitors.
Spokespeople for Barclays, ABN AMRO and BBVA declined to comment on Sunday.
TCI, which was part of a group of activist investors that torpedoed Deutsche Boerse's takeover bid for the London Stock Exchange in 2005, in February said ABN was significantly undervalued and called on it to merge, sell or spin off some of its assets or potentially the whole business.
Paul Kaju, a spokesman for TCI, was quoted on Monday as telling London's CityAM newspaper after news of the Barclays approach broke: "If a merger will unlock the group's undervalued assets then fine, yes, we would support that."