Lumley has impressed Bradbury and Cherries goalkeeping coach Neil Moss during a trial.
The 21-year-old is currently registered to Jerez Industrial through the Glenn Hoddle Academy's links with the Spanish club.
Lumley turned out for Cherries in a friendly at Reading last month and also featured in this week's practice match against the Hoddle Academy.
Cherries recalled Dan Thomas from a loan spell at Dorchester following the recent departure of Jon Stewart and Bradbury will be looking to bolster his resources for next season.
Bradbury said: Billy is going to train with us for the rest of the season so we can have a closer look at him. We need to bring in another keeper and he was recommended to Mossy.
We have liked what we have seen so far.
Lumley started his career in the youth ranks at Wolves before joining Hoddle's Academy in 2008.
Following an unsuccessful trial at Brentford, he eventually joined Northampton and made two league appearances.
Meanwhile, Bradbury and Joe Roach are next week set to make decisions on the futures of the club's second-year apprentices.
Bradbury has been monitoring the seven youngsters in training and in reserve games and will decide their respective fates following consultation with Roach.
The players concerned are Harry Bell, Paddy Hester, Shane Murphy, Alex Parsons, Tom Strong, Dan Strugnell and Matt Vokes.