Sunday’s 3-1 defeat against Manchester City means Chelsea have won just one of their last seven matches in all competitions.
Lampard turned on his Chelsea squad after a dispiriting 3-1 defeat to Arsenal at the Emirates on Boxing Day but he failed to get a response in their next fixture as the Blues were held by Aston Villa at Stamford Bridge before a resounding defeat to Pep Guardiola’s side.
Roman Abramovich is said to be considering sacking Lampard if the club’s poor results continue and former striker Andriy Shevchenko has been touted as a potential successor.
However, Tuchel’s departure from Paris Saint-Germain means a top-level manager is available on the market, while former Juventus boss Allegri has openly stated his desire to work in England.
According to the Telegraph, the pair are fighting to place themselves at the front of the queue should Chelsea decide to sack Lampard and both managers have made a play for the role at Stamford Bridge.
Lampard’s position is not thought to be under immediate threat but Abramovich has a reputation for sacking managers and the Russian will not countenance the club finishing outside the top four after a £220million spending spree last summer.
Only Ben Chilwell and Thiago Silva have impressed from Lampard’s summer arrivals so far this term, with question marks still hanging over Timo Werner, Kai Havertz and Edouard Mendy.
Hakim Ziyech has showed fleeting signs of what he’s capable of but the winger has struggled with injury since joining the Blues.