Both clubs have had a difficult start to the campaign with the Jam Tarts sitting 11th and their Capital rivals ninth after four league games.
It has led to a growing disgruntlement among both sets of fans as to the direction their respective club is heading.
Towards the end of Hearts' 2-2 draw with Hamilton on Saturday Craig Levein was implored to quit by the home support, while at Fir Park Paul Heckingbottom was on the end of receiving end of abuse from by the travelling fans as Hibs fell to a 3-0 defeat to Motherwell.
However, as revealed by The Scotsman, Budge, who herself has come in for criticism from the Tynecastle support, she has backed both sides to improve drastically and be fighting for a place in Europe come the end of the season.
“Normally we would be expected to be two clubs trying to achieve European football. At the moment, I have to say neither of us are sitting anywhere close to that.
“But we are only four games into the season, I have not given up hope yet. I should probably say: watch this space. I am confident both Edinburgh clubs will be fighting for a spot in Europe at the end of the season.”