Dyke, who is producing a creative industries review for the Conservative party, said that BBC programming is "in pretty good shape", but the corporation itself is a "bit short of friends".
Speaking to The Guardian, Dyke claimed that the way in which Thompson has handled the BBC's strategic reshuffle has failed its staff.
"On air, it's in pretty good shape but it's a bit short of friends. It didn't manage [Jonathan] Ross and [Russell] Brand very well. It doesn't have to pay its director general £800,000," he said.
"It is a good job, and Mark [Thompson] earns more than twice what I earned when I was doing it. The staff are whingeing. Mark is doing some great things, but he is not taking them with him."
Thompson today officially unveiled his strategic review of the BBC, titled Putting Quality First, which was leaked last Friday.
The report signalled an end to digital stations BBC 6 Music and Asian Network, as well recommending halving the BBC's website and reducing spending on sports rights and imported US programming.
Writing in The Guardian, Thompson said that the proposed changes are not "a piece of politics" or a sign that the BBC is retreating from digital platforms.
"The BBC should not attempt to do everything. It must listen to legitimate concerns from commercial media players more carefully than it has in the past, and act sooner to meet them. It needs the confidence and clarity to stop as well as to start doing things," he said.
"The proposed changes we are announcing are not a piece of politics - they are rooted in a clear vision of what the BBC exists to do. It is also not a blueprint for a small BBC, or a BBC that is in retreat from digital. That is the last thing the British public want."
Despite recent controversy and criticism, Thompson also expressed his belief that licence fee payers have not lost any faith or confidence in the BBC.
"I still believe that the central question for the BBC is: 'Can you maintain eye contact and ear contact with the British public?'" he said.
"And most encouragingly, there's no evidence at all of any erosion of public confidence in the BBC. And that's the very interesting thing about some of the 'crises' which envelop the BBC: they happen to some extent in a space which is different from the relationship with the public."
The BBC Trust and senior BBC management will ultimately decide which of the recommendations in Thompson's report should be put into action.