Economists polled by Reuters were expecting a quarter point cut in benchmark borrowing costs to 0.25%, which would be a new low in the Bank's 322-year history.
Mr Carney previously warned British families to prepare their finances for a three per cent rise in the coming years.
It was widely predicted that the Bank would raise interest rates following governor Mark Carney's remarks signalling monetary easing in the wake of Britain's decision to quit the European Union.
But the minutes suggested a rate cut may come next month, revealing that "most members of the Committee expected monetary policy to be loosened in August". He said an initial assessment would be published this week, while a full assessment and new forecast will be published with the August inflation report.
Expectations of a period of relative political and economic calm kept sterling on course for its best week since 2009 on Friday, although broader risks from last month's Brexit vote prevented a break past two-week highs above $1.34.
All Monetary Policy Committee members agreed to maintain the quantitative easing programme at £375bn, as expected, with Gertjan Vlieghe the only dissenting vote against standing pat on rates.
Decision to hold interest rates does not mean the United Kingdom economy is in the clear.
"The improved resilience of the core of the United Kingdom financial system and the flexibility of the regulatory framework, had allowed the impact of the referendum result to be dampened rather than amplified", according to minutes of the MPC meeting.
Ben Brettell, senior economist at Hargreaves Lansdown, said: "Initially August had looked more likely, but with economic data deteriorating and markets still nervous, it now looks probable the MPC will adjudge that immediate action is warranted".
Bank policymakers hinted at an expansion of that program, which is created to keep market interest rates low and encourage financial institutions to lend, is possible ahead of their next decision on August 4.
But Aberdeen Asset Management Economist Paul Diggle said the bank was taking its time to assess the full picture of the impact of the vote. "Taken together, these indicators suggest economic activity is likely to weaken in the near term".
Added to the swift appointment of new Prime Minister Theresa May and her cabinet, that points to a couple of weeks in which those betting on further weakness could be squeezed in a market already leaning massively against the pound.
Hargreaves Lansdown said the average instant access savings account had gone backwards in real terms since the reduction to 0.5%, falling behind the consumer price index of inflation by more than 12%.