The Scottish National Party leader made the accusation today while giving evidence at the Leveson Inquiry into media ethics.
However, The Observer's owner said that it had "been unable to find any evidence to substantiate the allegation".
Salmond mentioned the alleged accessing of his bank account when Robert Jay QC, counsel for the inquiry, asked if his phone had ever been hacked by journalists.
The first minister said that Strathclyde Police, who were investigating the hacking scandal in Scotland, had not contacted him to say he may have been a victim.
However, Salmond added: "What I can say is that I believe that my bank account was accessed by the Observer newspaper in 1999.
"My reason for believing that is I was informed by a former Observer journalist who gave me a fairly exact account of what was in my bank account that could only have been known to somebody who had seen it."
Elaborating, Salmond said that he had bought some children's toys from a shop called "Fun and Games", which the Observer had apparently hoped meant a different type of retail outlet.
"For example I bought some toys for my then young nieces in a toy shop in Linlithgow High Street which was called 'Fun and Games'," he told Lord Justice Leveson.
"The person who informed me told me this caused great anticipation and hope in the Observer investigation unit because they believed that perhaps 'Fun and Games' was more than a conventional toy shop."
The Guardian News & Media Group, which operates the Observer newspaper, said that Salmond had raised the issue of the alleged bank account access with the editor of the Sunday paper last year.
However, the company said that its own investigation had uncovered no evidence to support the claim.
"As we explained to [Salmond] last year, on the basis of the information he had given us, we have been unable to find any evidence to substantiate his allegation," said Guardian News & Media in a statement.
"As our response to him at the time made clear, we take this allegation very seriously and if he is able to provide us with any more information we will investigate further."
The Leveson Inquiry also heard today from deputy prime minister Nick Clegg, who said that he had met Rupert Murdoch on two occasions ahead of the 2010 election, but only exchanged a few sentences with the News Corp boss.