Official figures for the last quarter of 1998 will be released on Thursday. The BBC expects its Radio 4 audience to have pulled back above the crucial eight-million-a-week mark.
The corporation will use the data to argue that Mr Boyle's radically revamped schedule - which drew widespread criticism last year - is now succeeding.
Its own unofficial research shows that the number of listeners in October grew to 7.9 million from the 7.7 million per month for the third quarter of 1998. In November, the figure is thought to have increased by another 600,000.
Mr Boyle had said that if his schedule changes did not work he would resign as controller. However, he had never claimed that the changes were designed to increase listening figures in the short term.
A number of long-standing programmes were dropped in last April's revamp, including Sport on Four, Week Ending, and Kaleidoscope, with 53 new shows being introduced. This was coupled with a pounds 1m marketing campaign aimed at attracting a broader audience. However, 90 per cent of letters received by Feedback, Radio 4's programme for listeners' points of view, opposed the changes.
If Thursday's official figures - released by the Radio Authority Joint Audience Research (Rajar) - are not as good as expected, the BBC believes it will have a ready excuse: Radio 4's Test match cricket coverage during winter months often leads to a mass turn-off.
If the figures are good, the corporation will argue that previous audience research has been based on sample sizes that are too small to give an accurate picture.
"The problem has been that the research takes individual snapshots of different listeners and we get these big swings," said a BBC insider. "The radio industry's research method is being changed later this year to give more continuity. That should iron out some of the anomalies such as Diana's death or the World Cup, which can have a bigger impact on the research than they actually have on listening."
Listening figures for the Today programme are known to have increased over the past three months as it has settled into its longer format.
Radio 4 is also confident that its post-9am programmes have been stronger after a number of changes, including the replacement of Matthew Parris's Mothers and Sons.
Further tinkering with the schedule this week will see the replacement of some of the lunch-time quiz programmes, which the station has admitted are not working.
From tomorrow, the musical quiz Full Orchestra is being replaced by a music feature every Tuesday.
From Thursday, the struggling antiques quiz Hidden Treasures will give way to a repeat of the rural affairs programme Open Country.
Radio, Review, page 17