There has been no order preventing Barnea or the Mossad from returning to Qatar to restart hostage negotiations.
The Hostages and Missing Families Forum said that they feel like the government is playing "Russian roulette" with the hostages.
Recent developments are drawing greater attention to the agency's role in the region.
Every day, Barnea will receive a list with the names of the hostages who will be released the following day, according to N12.
The outcome of the talks between the American and Israeli intelligence agency heads with the Qatari Prime Minister Sheikh Mohammed Bin Abdulrahman al-Thani was unclear.
The flight is suspected to have been part of ongoing efforts outside the public view to negotiate for the release of Israeli hostages currently held captive by Hamas in the Gaza Strip.
The details, and any potential results, of Barnea and Cohen's trip to Qatar have not yet been revealed.
"[This is] totally fake news," the Mossad statement explained. "The director of the Mossad did not meet with Shikma Bressler.
Without the backing of key figures in the defense establishment, Strategic Affairs Minister Ron Dermer will struggle to advance a US-Israel defense alliance.
The director of Mossad is helping frame the trilateral Israel-Saudi Arabia-US deal. Here is how.