The towering 20-year-old has won only six caps and scored just once but will surpass former United players Denis Law and Gordon McQueen, who captained Scotland at the age of 22.
Fresh from helping United lift the FA Cup after a 3-0 win over Millwall last Saturday, Fletcher told reporters in Tallinn: "Winning the FA Cup was a special moment for me and this has completed a great week. To captain Scotland has always been a boyhood dream and I feel very proud and honoured."
Coach Berti Vogts said Fletcher is his first choice in the absence of West Ham United defender Christian Dailly, who is preparing for Saturday's first division playoff final with Crystal Palace, and injured Blackburn Rovers midfielder Barry Ferguson.
"He (Fletcher) is a special boy and he is the future captain of Scotland," said former Germany manager Vogts. "Darren is one of the guys who has a key position in my team.
"He has cold water in his veins, he is so strong and has a strong personality. He is one of the biggest talents in Europe."
Vogts has tipped Fletcher to succeed Roy Keane as United captain and he is already one of Scotland's best prospects.
He came off the substitutes' bench to secure his country's place in the Euro 2004 playoffs last October with the winner in a 1-0 victory over Lithuania.
Fletcher was then pivotal in the surprise 1-0 win over the Netherlands in the playoff first leg a month later before the Dutch found their form to coast through 6-0 in the return.
Barring injury, Fletcher is also likely to captain Scotland in Sunday's home friendly with Trinidad and Tobago in Edinburgh.
