Robinson spent the second half of last season on loan with the Baggies but efforts to bring him back on a permanent basis have so far been held up by the Blades demanding an £8million fee.
Burke could now break the logjam with Chris Wilder believing he can bring the best out of the Scotland international after a miserable three years at The Hawthorns.
The 23-year-old has started only two league matches since joining from RB Leipzig for £15million and has spent the last two seasons out on loan.
Talks are now progressing over the specifics of a deal which could end up either being a straight swap or separate transfers for each player.
Robinson joined Albion on loan in January and went on to play a key role in the club’s promotion to the Premier League, scoring three goals in 16 appearances.
Able to play on the left-wing and up front, Slaven Bilic is eager to sign the 25-year-old on a permanent deal but with finances tight, the Baggies are reluctant to pay what Sheffield United were asking for the Republic of Ireland international.
Robinson moved to Bramall Lane from Preston North End in July last year for a fee in the region of £8million and the Blades have been keen to recoup that money.
Yet Albion have made it clear they are willing to pay £4million and talks were in danger in stalling before Wilder’s interest in Burke emerged.
It is understood Robinson is keen to join Albion on a permanent basis but also willing to stay and fight for his spot at Sheffield United if necessary – and determined to prove he is good enough for the Premier League.