A BBC spokeswoman said the presenter regrets the “ill-advised” conversation.
A source at the BBC said management were “deeply unimpressed” with the unaired discussion, which is reported to have taken place before Monday morning’s edition of Today.
Speaking in the Radio 4 studio, Mr Humphrys reportedly asked Mr Sopel about “how much of your salary you are prepared to hand over to Carrie Gracie to keep her”.
He then referred to “other men who are earning too much” at the BBC.
Mr Sopel is understood to have replied that “if we are talking about the scope for the greatest redistribution I’ll have to come back and say well yes Mr Humphrys”.
The presenter is then reported to have uttered a profanity and said that he was “still left with more [pay] than anybody else”.
Miriam O’Reilly, who won an ageism case against the BBC in 2011 after being dropped from Countryfile, described the exchange as “base, smug and condescending”.
Claiming to have heard a recording of the chat, Ms O’Reilly said it represented the attitude of “back-slapping entitled males”.
A BBC spokeswoman said: “This was an ill-advised off-air conversation which the presenter regrets.
“The BBC is committed to getting its pay structures right and, as we have said, we are conducting a comprehensive analysis of presenter pay,” she added.