Bilic continues to face serious questions about his future after the Hammers' disappointing start to the season - but their most recent performance undoubtedly lifted some of the gloom.
West Ham booked their place in the quarter-final of the Carabao Cup after going 2-0 down to Tottenham at Wembley in midweek, as they staged a stunning comeback to score three times in the second-half.
The Mail reports however, that Bilic is not out of the woods yet, and the owners want more proof that their poor run is out of their system - starting with their crucial Premier League clash with Crystal Palace this weekend - before they decide to keep faith with the Croatian.
Should they decide to make a change, the Mail reports that Burnley boss Dyche is one name that the West Ham are seriously keen on after the fantastic job that he has done at Turf Moor.
And, perhaps ominously for Bilic, the Mail adds that West Ham have worked out how much they would need to pay to get Dyche out of his contract with the Clarets, and how much they would be able to pay him, which certainly hints that their level of interest is edging past the point of being tentative.