The midfielder has been strongly linked with a move to Fratton Park from Croatian outfit Dinamo Zagreb this week.
But The News understands the 21-year-old is no longer at the club after spending a period on trial.
Burton arrived last week and played in the warm-up defeat against Chelsea under-23s at Cobham last Tuesday.
The former Arsenal man produced a competent showing, including going close with an acrobatic effort in the first half.
He was withdrawn at the break before coming back into tthe fray after the restart in the 2-0 defeat.
But after Cowley ran the rule over the prospect who moved to Dinamo for £800,000 last year, he was allowed to leave PO4.
Pompey boosted their ranks this week with two signings, as striker George Hirst signed on loan for the season from Leicester after free agent Connor Ogilvie arrived following his Gillingham exit.
Cowley is still extremely light in the middle of the park, however, with just veteran Shaun Williams and Ryan Tunnicliffe his central options.
An injury to either of those men as the season begins at Fleetwood this weekend would present an early-season selection crisis for the 42-year-old to contend with.
The double addition this week takes Cowley’s signings to 10 this summer, with Gavin Bazunu, Liam Vincent, Connor Ogilvie, Clark Robertson, Jayden Reid, Gassan Ahadme, Williams and Tunnicliffe.
That gives Cowley a squad off 22 players, although new recruit Reid is sidelined long term after injuring his knee in the pre-season clash with Luton Town.