According to the WCC, Kirill expressed a "serious concern" about some of the challenges facing ecumenical dialogue in view of what he termed the "new positions of some Protestant churches" on several important moral issues, including their understanding of homosexuality.
The Episcopal Church in the USA in particular has pursued a liberal agenda in the last decade, consecrating its first openly gay bishop in 2003 and consecrating its first partnered lesbian bishop this year. The actions have angered traditionalist Anglicans worldwide and met with alarm from other denominations, including the Orthodox and Roman Catholic Churches.
Dr Tveit said the discussion with the Patriarch had been "frank".
He said: "His Holiness challenged the council to articulate how it can contribute to build bridges in this situation."
In spite of this, Kirill spoke of his "deep commitment" to the WCC. Dr Tveit praised the work of the Russian Orthodox Church in reaching out to youths.
"The Russian Orthodox Church can play an inspiring role in the fellowship of WCC member churches", he said.
"Bringing the younger generations to the faith is a common concern for all of our member churches.
"We have learnt about some of the initiatives taken in this regard under Kirill's leadership and I think all WCC member churches could benefit from them."
On the last day of his visit today, Dr Tveit will meet the Very Rev Arkady Shatov, chairman of the Department for Church Charity and Social Services, a visit to the St Dimitry School of Sisters of Charity, and meetings with Russian government officials.
The Russian Orthodox Church joined the WCC in 1961 and is its largest member, with five of its representatives currently sitting on the WCC's 150-member Central Committee.
Joining him on the visit is the WCC deputy general secretary Georges Lemopoulos, as well as WCC staff members, the Rev Dr Dagmar Heller (Ecumenical Institute Bossey/Faith and Order) and Fr Dr Daniel Buda (Church and Ecumenical Relations).
Following the visit, the Russian Orthodox Church will host the annual meeting of the Permanent Committee on Consensus and Collaboration, which has the responsibility for continuing the dynamic of the Special Commission on Orthodox Participation in the WCC.