Boro managed a surprise 2-1 win over Aston Villa on Sunday.
“There are no targets. It is about one game at a time. We know that we need to play well at home. We haven’t done that too well,” said Downing.
“Everton will be a tough game. They are good at home but, if we can pick something up there, we will be flying and hopefully will go on a good run.
“Europe is definitely possible. We’ve had experience of it in the past. That is my aim and the aim of the players. I think they can sense it a little bit as well.
“They have had a couple of good results and are in and around there and, if we can come to places like Villa and win, there is no reason why we can’t go anywhere else and win.
“Winning at Villa gives you a lot of confidence. They were fifth in the league going into the game and they hardly ever lose at home.
“We knew what their strengths were. They are good on the counter-attack but I thought the lads at the back did really well and allowed us to get forward even more.
“I thought going forward we looked a good team. It was good finishing by Tuncay and we could have had more. There were a lot of attacking players on show from both sides but we got the result we deserved.
“Now it will be nice to get something at Everton as well and build on it. Obviously if you lose a couple of games, you are back down there again and it is a big game for us.
“It is a very tight league. A couple of wins and you are flying. A couple of losses and it changes so we have to keep picking points up.”
