The Blues have had an initial bid rejected by Peterborough for the prolific 27-year-old.
It is believed to be around the £425,000 mark – the amount the Posh splashed out to recruit him from Stevenage 12 months ago.
There are indications boss Darren Ferguson is still prepared to do business, following the arrival of fellow striker and ex-Pompey target Mo Eisa.
However, the Blues would be required to significantly increase their offer to capture a player who netted 18 times during his first full League One season.
Yet interest from Fratton Park still exists as Jackett prepares to ramp up his summer recruitment.
Increased striking options are essential, with just Oli Hawkins and Brett Pitman on the Blues’ books, of which the latter is preferred in a number 10 role.
Jackett is seeking more variety among his frontline players, bringing Godden to his attention.
Although Pompey finished as League One’s second-highest scorers behind Luton, their remains scope for bolstering their striking pool.
The former Scunthorpe trainee possesses an impressive scoring record since returning to the Football League three seasons ago, following time spent in the non-league game.
Snapped up by League Two Stevenage from Ebbsfleet in June 2016, he netted 21 times during that maiden campaign.
It includes registering in both games against Paul Cook’s Pompey, as the Boro inflicted the double over the eventual League Two champions.
During his second season, Godden’s return of 14 goals took his tally to 35 in 87 appearances for the club, prompting his sale to Peterborough in June 2018.
Last season he netted the Posh's consolation in their 2-1 home defeat to Pompey in September 2018, which lifted the visitors to the top of League One.
It represented the third-straight occasion he had scored against the Blues in the Football League.
Godden also faced Pompey during their 2014 pre-season schedule, appearing as a substitute for Ebbsfleet in a friendly.
Andy Awford’s men suffered a shock 3-0 defeat at Stonebridge Road, with the striker replacing two-goal Charlie Sheringham in the 76th minute.
Posh manager Ferguson is seeking to partner £1m capture Eisa with Ivan Toney during the 2019-20 campaign.
That will reduce Godden to the role of back-up, despite 18 goals in 47 outings last term.
Ferguson, though, may be loath to sell to a club which represent promotion rivals in the battle to reach the Championship.
Nonetheless, Jackett is in the market to strengthen his attack, with Ipswich's Ellis Harrison another on his radar.
And Godden also continues to interest the Blues ahead of the return to pre-season training on Thursday, June 27.