Dutchman Koeman guided Barca to a third-placed finish in LaLiga this term, meaning they finished outside of the top two for the first time since 2007-08, though the Blaugrana did succeed in winning the Copa del Rey.
They were inspired by Lionel Messi, who scored 30 league goals to win an eighth Pichichi trophy, though the 33-year-old is also the subject of speculation over his next move, with his contract at Barca up next month.
Laporta succeeded previous president Josep Maria Bartomeu in March, taking charge at Camp Nou for a second spell.
Koeman was appointed by Bartomeu last year, after Quique Setien's ill-fated tenure, with the former Netherlands coach signing a two-year contract.
Reports have suggested that Laporta is keen to install his own man in the hot-seat, and on Tuesday he met with Koeman and the Barca legend's agent.
No firm decision was made, according to reports in the Spanish media, though Marca suggested there was a possibility of another meeting later in the evening.
After a poor start to the congested season, Barca did manage to drag themselves into title contention, only to slip up towards the end of the campaign.
Their tally of 79 points is their lowest total since 2007-08, when they managed 67. Indeed, the last coach to earn 76 or fewer points in his first 38 games with the club was Frank Rijkaard in 2003-04 (72 - W21 D9 L8).
With Atletico Madrid triumphing this year and Real Madrid winning LaLiga in 2019-20, it meant Barca have failed to win the title in two successive seasons for the first time since 2008.
Uncertainty over Koeman's future has not stopped transfer links, although – in large part due to the impact of the COVID-19 pandemic – Barca's finances have been a cause for concern.
The majority of the first-team squad are reportedly up for sale should offers come in, though the Blaugrana are seemingly in the market for free agents.
Sergio Aguero looks set to join once his contract at Manchester City expires, while Memphis Depay and Georginio Wijnaldum – Netherlands regulars under Koeman – and Eric Garcia also look likely to arrive.
Barca's issues were compounded on Tuesday, when UEFA confirmed it had opened disciplinary proceedings against the club, along with Real Madrid and Juventus, for their part in the failed European Super League project, with none of the three teams having yet withdrawn.