Moyes's side have endured a terrible start to the New Year but they did manage four wins, a draw and two defeats from seven league matches at the end of 2013.
Chelsea had five wins from seven games, one draw and a defeat while Martinez, who replaced Moyes at Everton, led his side to four wins from six matches, one draw and one defeat.
United started December with a 2-2 draw against Tottenham before subsequent 1-0 home defeats to Everton and Newcastle.
A run of four wins followed, against Aston Villa, West Ham, Hull and Norwich, before United lost 2-1 at home to Tottenham on New Year's Day.
Swansea knocked United out of the FA Cup on Sunday before Sunderland beat them 2-1 in the first leg of their League Cup semi-final.
It is the first time since 2001 that United have lost three games in a row.