Ramsey secured a place in the club's history books when he scored the winner in last season's 3-2 FA Cup final triumph over Hull City and told reporters on the eve of Saturday's semi-final against Reading that it had only whetted his appetite for more success.
That is more likely to come by retaining the FA Cup than winning a first league title since 2004 with Chelsea seven points clear of Arsenal at the top of the table, having played one fewer match.
But if Chelsea lose at home to Manchester United on Saturday and to Arsenal, who have won their last eight league matches, at the Emirates next weekend, the outcome might not look quite so clear-cut.
"The double? Yes, that is still a possibility," said the 24-year-old Welsh international.
"We are just trying to win our remaining games. We are on a fantastic winning run and that’s all we want to keep on doing.
"It’s going to be difficult. Chelsea have still got a lot of points so they will have to slip up a few times for us to catch them so it will be difficult -- but it's still possible."
Arsenal have won the double three times, in 1971, 1998 and 2002. On the first two occasions they only clinched the title by a point.
"You cannot rule it out," said Ramsey. "When you have that momentum it is still possible. Even though it will be difficult we have to win our remaining games and see where that leaves us."
Even if they fail Ramsey believes Arsenal's long wait for the title will soon be over.
"I feel we're getting closer," he said. "Even last season we started off really well and maybe just faded away at the end.
"We lost against the bigger teams but this season we have done a bit better against them.
"This season we didn’t start too well, it was a bit interrupted, but the second half of the season we have been really good and consistent in our performances. I feel we are very close."
The winners of the Arsenal v Reading game will play Liverpool or Aston Villa in the final at Wembley on May 30.
(Editing by Ed Osmond)
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