The FA has awarded the 22nd minute goal - the 2,000th England have scored since they began playing internationals in 1872 - to Manchester City midfielder Gareth Barry.
However his header took a deflection off the head of Sweden defender Daniel Majstorovic leaving some organisations giving the goal to Barry and others to the Swede.
England's 1-0 win represented another landmark for the home side - their first win over Sweden in 43 years.
An FA spokesman said: "We have awarded the goal to Barry. However, if the referee, in his report, states the goal was an own goal, then we will change it in line with his report.
"We will ask him to study the footage and take a view. If the match would have been in a FIFA or UEFA competition, the competitions committee would decide, but in a friendly, it is the referee's decision."
The Wembley scoreboard displayed Barry's name as the scorer for the rest of the match but England manager Fabio Capello said he was not concerned with who scored the landmark goal.
"It is just statistics, for me it is not important. It might be important in 10 years who scored the 2000th goal, but not now," said the Italian.
British bookmakers William Hill are paying out to any punter who backed Barry as being the first, last or anytime scorer and also on anyone who backed a 1-0 scoreline with Barry scoring.
"The scoreboard at Wembley displayed throughout the game that Gareth Barry was the goal-scorer but despite Daniel Majstorovic being credited as having got the final touch we are celebrating a landmark 2,000th England goal by paying out to any punter who backed the City midfielder to score," said Hill's spokesman Joe Crilly.
England's first goal was scored by William Kenyon-Slaney in a 4-2 win over Scotland on March 28, 1873 and the 1000th goal came from Jimmy Greaves in a 5-1 win over Wales at Wembley on November 23 1960.
