In a tweet, Foreign Minister Mevlut Cavusoglu called the team’s treatment “unacceptable in terms of diplomatic and humanitarian practice” and said Turkey would “do what is necessary”.
As journalists surrounded some players on arrival a dish-scouring brush could be seen among the microphones.
Players also endured a lengthy customs search, Turkish media reported.
According to Turkish broadcaster NTV, Turkey has delivered a diplomatic protest to Iceland.
Iceland’s ministry of foreign affairs said that a request for “fast-track security” from the Turkish embassy was sent only hours before the team’s arrival, and so was never processed.
But in any event, “such privilege is usually only accorded to senior government officials”, it said.
Since Turkey is not in the European Union or Schengen zone, which has common security rules with Iceland, standard procedures applied – which for the entire Turkish team and staff took one hour and 23 minutes, the ministry said.
The Turkish team arrived on Sunday, ahead of a match against Iceland on Tuesday in the Uefa Euro 2020 qualifying competition.
Turkey currently tops Group H, having won all three of its matches so far in the six-nation group. The team beat world champions France 2-0 on Saturday.
Burak Yilmaz, a striker for the Turkish club Besiktas, was quoted as saying the team were kept waiting at Icelandic customs for three hours – far longer than the Icelandic officials reported – during a painstaking bag search.
Turkey’s President Recep Tayyip Erdogan is a football fan, having played semi-professionally before he was elected as mayor of Istanbul in 1994.
His spokesperson Ibrahim Kalin tweeted to say that “disrespect to our national team in Iceland is unacceptable”.
“Our state and nation stands next to the national team, who will give the best answer on the field,” he said.
President Erdogan was best man at the wedding of the German footballer Mesut Ozil, who is ethnically Turkish, on Friday.
There was also diplomatic tension surrounding the Turkey-France match on Saturday in Konya.
French President Emmanuel Macron said sustained whistling by Turkish fans while the French national anthem was being played was “unacceptable”.
On Monday Noël Le Graët, head of the French Football Federation, said the whistling was “regrettable” but added: “for me, it does not amount to an incident”. His response suggests that France will not escalate the matter with Turkey.
President Macron is among the most vocal opponents in the EU to Turkey’s bid to join the 28-nation bloc. Turkey’s talks on EU membership are currently deadlocked.