The Leith club lost 9-0 on aggregate to Swedish side Malmo in the Europa League and have now been defeated in their two opening league matches.
Callum Paterson headed home the only goal in the 71st minute to end Hibs' five-game unbeaten run against their rivals.
Fenlon said: "I've got no problem with people looking at my position, no problem at all. I'm the manager of the club and I will continue to do that.
"We've lost two games 1-0 in the league and Europe was really disappointing. In both league games there was very little in them and I don't think we're too far away from getting a result. I'm very happy with the squad we have.
"We need one result, footballers live confidence-wise and a result on the board helps, it breeds confidence.
"There has been very little in the games we played and with a break of the ball, we could have got a result."