Everton's form has improved significantly since David Unsworth's penultimate game in caretaker charge. After beating West Ham with Big Sam looking on before his appointment was confirmed, the club have won four out of five games since he officially took charge, conceding just two goals in that time.
Nevertheless, The Sun suggests that Allardyce is keen to add to his options during next month's window, and has identified Crystal Palace ace Tomkins as a potential target.
It has been a difficult season for the 28-year-old, with the Eagles spending much of the early part of the campaign bottom of the Premier League table and with their survival hopes being written off by many.
However, Roy Hodgson's men have turned it around in impressive fashion lately, and Tomkins has featured in their last three games, during which they have taken seven points and kept a clean sheet away at Leicester City last weekend.
Perhaps unsurprisingly, The Sun reports that Palace would not want to sell the centre-back in January as they still sit in a precarious position in the top flight - and thus, Everton may have pay a sizeable fee if they hope to get a move across the line.
It will be interesting to see whether Allardyce is willing to give a pursuit the green light given that Tomkins is not the youngest player and has perhaps not quite kicked on as well as many expected when he was coming through the ranks at West Ham.
But with Big Sam knowing exactly what he will get from the defender having worked with him for a number of years at the Boleyn Ground, perhaps he may well be tempted to ask the board to meet Palace's demands next month.