Ogbene, who has also played for Cork, Limerick and Brentford, joined Luton from Rotherham on a free transfer last summer and impressed in the Premier League, despite the Hatters’ relegation.
He scored five goals in 39 games in all competitions and was the second fastest top-flight player last season, after Tottenham’s Micky van de Ven, having clocked 36.93kph against Fulham last September.
Ipswich, who travel to AFC Wimbledon in the Carabao Cup on Wednesday, also remain in talks with Chelsea for striker Armando Broja ahead of Friday’s transfer deadline.
McKenna said: “We’ve spoken to him, I think that’s well known. There’s ongoing discussions with Chelsea but I’ve not been privy to them over the last 24 hours. We’ll see if it’s the right thing for the club and the player.”