He admitted he is interested in moving into football management after being second favourite to becoming Martin Scott's successor at Victoria Park.
The 45-year-old is after his coaching badges, and he has also worked as a coach in the England set-up under Kevin Keegan.
He ended his playing career at Hartlepool in 1999 after two spells at St James' Park.
Beardsley: "I have a great affinity with Hartlepool United. I thoroughly enjoyed my time there and it is a great club. The fans are very passionate and they have a great chairman in Ken Hodcroft. I had a great relationship with the chairman when I was there, he does things by the book and never goes back on any promises he makes.
"The progress they have made in recent years is unbelievable and it is testament to the work of the chairman that they are now an established League One side. I still have an important job at Newcastle United in their academy that I enjoy very much. But I must admit that management is something that appeals to me and I am coming to the end of my coaching badges and I am not far off considering a move into management.
"The Hartlepool job would be a fantastic job for whoever gets it. There has been a lot of names mentioned for it, I have heard mine's included, and the job (caretaker boss) Paul Stephenson has done must put him in the frame, although I believe he has ruled himself out of it."