The proportion of households expecting a rate rise within 12 months sank to 41 percent in April from 46 percent in March, the lowest proportion since January.
The share of households who said they did not know when the BoE would raise rates rose to 29 percent from 23 percent.
The survey was conducted from April 11-14, before the publication of data last week showing a big drop in the unemployment rate to 6.9 percent and prompting financial markets to bring forward expectations of a rate rise to March 2015.
The BoE has said repeatedly that it is in no hurry to raise rates and wants to see the slack in Britain's economy largely used up first.
"The lower interest rate expectations can be seen as a sign that forward guidance is working successfully," said Markit economist Chris Williamson. "However, the fact that inflation also continues to fall ... will have no doubt also contributed to the lowering of rate hike views," he added.