Since Westley, 48, took charge in October the Exiles' form has improved.
Newport are eight points off the play-off places, but Westley has issued his players with an ambitious challenge.
"We can look at sixth place and think two or three weeks down the line it's realistic we can put ourselves in contention," said Westley.
"That will be a nice starting point. If we can get to Christmas time and we are in the mix, that's a nice point to be at.
"I've won promotion out of this league from 18th in January, so we all know what's possible."
The Exiles are unbeaten in their last six games, a sequence that has seen them beat AFC Wimbledon in the Checkatrade Trophy and progress to the second round of the FA Cup after a replay against Alfreton.
In the league there has been a draw with Barnet, victory over Accrington Stanley and an impressive three points taken from Carlisle, who were unbeaten at the time.
But, with his sights set so high and a trip to eighth-placed Notts County coming on Saturday, Westley insists keeping the current run going will not be enough to satisfy him.
"We want to improve what we are doing, that's the key," he added.
"If you get into the frame of mind where you want to keep things going, you've lost the plot and are on the road backwards. The route forward has to be improvement.
"We've got a lot of work to do, we don't underestimate it, right now we are producing decent form, but we have to produce that form on a sustained basis if we are going to mount any sort of campaign."
Newport County manager Graham Westley wants his side in the promotion race by Christmas, despite them currently propping up League Two.