Indeed the England manager intends to convince Carragher - who has collected just five of his 34 England caps under McClaren's regime - that, at 29, he can be a key member of the national squad for several years to come.
However, Carragher, who regularly featured under Sven-Goran Eriksson, appears to be tiring of life on McClaren's bench. A source close to the player said: "Jamie does not want to sound like a big-time Charlie but he just feels that at this stage of his career it would be better if he concentrated his efforts on helping Liverpool become the No1 team in England again. He has always regarded it as a tremendous honour to represent his country and he has happy memories of pulling on the shirt. But things have to end some time and he thinks this might be a good time to bow out."
McClaren, whose man-management skills may be fully tested when he meets Carragher for discussions in the near future, is determined to dissuade him from following Manchester United's Paul Scholes into international retirement.
Carragher is in some ways a victim of his own versatility. He can play across the defence and as a holding midfielder but prefers to operate as an orthodox centre-half. For that position, however, he must compete with John Terry, Rio Ferdinand, Jonathan Woodgate and Ledley King.
The Liverpool defender was seemingly left hugely disappointed when he was named as a substitute for England's Euro 2008 qualifier in Estonia last month with King preferred in McClaren's starting XI.
Nevertheless McClaren is understood to rate Carragher's defensive abilities highly. Moreover he is well aware that this most popular, down-to-earth character is both low-maintenance and a great traveller - qualities which can prove invaluable amid the stress of tournament football.
Carragher is one of the principal successes to emerge from the Football Association's former National School of Excellence at Lilleshall. His engaging personality led Peter Taylor and Howard Wilkinson to name him as a personal favourite during their respective tenures in charge of the England Under-21 side.
Wilkinson once said: "Jamie is a bit of a slow burner who will take his time to get into the senior squad but, once there, he'll stay there. He's exactly the sort of player England need, both on and off the field; you need people like Jamie Carragher at big tournaments."
Carragher did not play yesterday as Liverpool enjoyed a victory in their first friendly of the summer, beating Wrexham 3-2 in their annual meeting.
The Austrian teenager Besian Idrizaj scored an impressive 19-minute hat-trick as Rafael Benítez's squad of promising youngsters opened Liverpool's pre-season in some style at Wrexham's Racecourse Ground, where a crowd of 11,210 watched both sides field completely different teams in each half.
Liverpool's 22 players comprised the bulk of their FA Youth Cup-winning sides of the past couple of seasons. The only big names on show for the visitors were Steve Finnan, Mohamed Sissoko, Alvaro Arbeloa and Jermaine Pennant.
Pennant delivered three crosses for Idrizaj to give Liverpool a 3-0 lead before Eifion Williams scored Wrexham's goals.