Bedene, who started the year at No 145 and has risen to world No 83, has been a resident in the UK since 2008.
The 25-year-old has made it known that he is keen to play in the Davis Cup for Great Britain and is planning to appeal against a new rule which prevents players from representing more than one nation.
Bedene, who lost in the final of the Chennai Open to Stan Wawrinka in January, has represented Slovenia in the Davis Cup on three occasions but will appeal on the grounds his passport application was lodged before the rule change came in.
British No 1 Murray feels Bedene, who reached a career-high ranking of 71 in 2013, could prove to be a very useful addition to Leon Smith’s side.
But he admits Bedene will find it difficult to break into the current line-up, which reached the quarter-finals of the competition courtesy of a win against the United States in Glasgow earlier this month.
“I don’t know him extremely well but I know that he’s obviously playing some very good tennis right now,” Murray told Sky Sports after his victory over Donald Young at the Miami Open on Friday.
“I think maybe in the future it’s possible for him to be involved in the Davis Cup team but I’m not sure exactly what the rules are.
“I know the rules change a little bit this year, that you’re not allowed to play for two countries, so I’m not 100 per cent sure if he’s going to be eligible.
“This year we have a team that’s being doing extremely well and I would expect that this year we would stick with that team and revisit it again in the future if he becomes eligible.”