After being treated by medical staff, Ospina was then able to continue playing with bandaging around his head.
But later in the first half he collapsed and was stretchered off in the 44th minute.
Napoli had gone 2-0 up in the game before conceding twice in six minutes as Udinese levelled the match before a lengthy stoppage for Ospina to receive treatment.
The keeper was reported to have lost consciousness for a few minutes.
He did regain his consciousness and was taken by an ambulance to hospital where he will be closely monitored for a number of hours.
Ospina departed the Emirates Stadium on a season-long loan deal last summer after requesting a temporary move following the arrival of Bernd Leno from Bayer Leverkusen.
The 30-year-old shot-stopper has started 21 times for Napoli since that move and has competed with Alex Meret, who replaced him against Udinese, for the number-one spot.
His efforts have helped the Naples club to second place in Serie A this term, 15 points back from league-leaders Juventus.
And Ospina's collapse has come at a time when debate is rife around head injuries in football and sport as a whole.
Rules around the treatment of concussion have been tightened considerably in recent years with.
In football, some people have linked heading the ball with brain trauma - a theory that gained ground when an inquest into the death of former footballer Jeff Astle ruled he had died due to heading heavy leather footballs.
Astle died in 2002 at the age of 59 and suffered with Alzheimer's disease for nearly 10 years.