Threadneedle Street issued the statement after its governor Mark Carney yesterday said it is making contingency plans in the event of an independent Scotland.
Mr Swinney had earlier in the week said that Holyrood ministers had had "technical discussions with the Bank of England regarding our proposal for a currency union".
Better Together campaign leader Alistair Darling said it was a 'rebuke' for the SNP.
The Bank's statement said it had noted Mr Swinney's comments, "regarding technical discussions between officials of the Scottish Government and the Bank of England" about currency union.
It added: "To be clear, consistent with its statement in December 2012, the Bank of England has not entered into discussions with representatives of the Scottish Government about proposals for future monetary arrangements in Scotland.
"As the Governor said yesterday, the design of any changes to UK monetary and financial arrangements would ultimately be a matter for negotiation between the Westminster and Scottish Parliaments. The Bank of England will deliver whatever remit it is given."
On Wednesday, Mr Swinney had said: "The Scottish Government has had technical discussions with the Bank of England regarding our proposal for a currency union and we welcome their continued acknowledgement that the bank will introduce whatever the politicians decide."
Better Together leader Alistair Darling said: "I cannot remember a time when the Bank of England have had to rebuke a politician in such stringent terms.
"Yesterday John Swinney told us the Scottish Government was having discussions with the Bank of England on a currency union. Today the Bank of England have said categorically that no such discussions on future monetary arrangements took place."
Scottish Conservative leader Ruth Davidson said: "It is utterly embarrassing that Scotland's Finance Secretary has had to be slapped down by the Bank in this humiliating manner.
"It is now clear there are no discussions between the Bank and the Scottish Government on future monetary arrangements and there never have been."
Scottish Liberal Democrats leader Willie Rennie added: "The Nationalists have repeatedly sought to inflate the credibility of their currency plans. It is very serious that the Bank of England has taken the unusual step of correcting the statement from John Swinney. We need John Swinney to explain why he exaggerated.
"He must now provide complete clarity over those claims and be upfront with the people of Scotland."
A government spokesman insisted Mr Swinney's statement had accurately reflected the nature of the conversations with the bank's officials.
He said: "The Scottish Government has had ongoing technical discussions with the Bank of England, as the Bank itself has previously confirmed.
"These technical discussions were inaugurated by former governor Mervyn King and continued under the current governor, Mark Carney, as confirmed in the statement issued after the meeting between the First Minister and Mr Carney on January 29 this year, which was agreed by the Bank.
"We have always made clear that these are technical discussions and not negotiations, and that we respect the neutrality and independence of the Bank, which had made clear it will implement whatever is agreed."