Mr Meshaal himself had indicated last year that he would not be seeking a new term.
"We are forging ahead with the reconciliation, " Mr Meshaal told the BBC's Hardtalk programme.
Jordan closed the bureau in the late 1990s and briefly jailed Mr Meshaal before expelling him to Qatar.
Carter also asked Hamas to release Shalit, Meshaal said after the former president's visit, but the request was rejected.
Former President Jimmy Carter held a controversial meeting with the group's leader, Khaled Meshaal, in April of this year.
Mr Meshaal then headed to Syria, where he became political leader in exile.
Mr Meshaal also said preparations were being made for presidential and parliamentary elections.
But, in an official statement, Hamas urged Mr Meshaal - who lives in exile in Syria - to reconsider his decision.
Correction: This report has been amended to remove a line saying that Khaled Meshaal said President Assad continued to support Hamas.
Mr Abbas and Hamas leader Khaled Meshaal signed a reconciliation deal in Cairo in 2011, but it has not been implemented.
BBC: Gaza: Fatah holds biggest rally since Hamas came to power
In a speech during a visit to Gaza last month, Mr Meshaal urged "reconciliation and national unity of the Palestinian ranks".
BBC: Gaza: Fatah holds biggest rally since Hamas came to power
Last week, Mr Mursi held talks with Hamas's political leader, Khaled Meshaal, and Palestinian Authority President Mahmoud Abbas, who heads Fatah.
BBC: Egypt's president discusses Gaza blockade with Hamas PM
Meshaal -- who lives in exile in Syria -- is the head of Hamas, which rules the Palestinian territory of Gaza.
The agreement followed talks, the first in 18 months, between Palestinian leader Mahmoud Abbas of Fatah and exiled Hamas leader Khaled Meshaal.
But in February 2012, Mr Meshaal and the rest of the political leadership in exile moved from Syria to Egypt and Qatar.
The political leader of the Palestinian Islamist movement Hamas, Khaled Meshaal, does not want to stand for re-election this year, the movement says.
Morsy said he invited Palestinian Authority President Mahmoud Abbas to meet with Khaled Meshaal, the political leader of Hamas, who lives in Cairo.
Our correspondent says Mr Meshaal's decision will please those in Gaza who see themselves as more hardline and committed to the struggle with Israel.
In 2011, Mr Meshaal and Palestinian Authority President Mahmoud Abbas - the Fatah leader - endorsed an Egyptian plan to reconcile the rival factions.
Mr Meshaal this year strongly supported Mr Abbas's move in upgrading Palestinian status at the United Nations to that of a non-member "observer state".
Egyptian President Mohamed Morsy has also invited Abbas and Hamas political leader Khaled Meshaal, who lives in Cairo, to a meeting to discuss Palestinian unity.
It is not clear who might replace Mr Meshaal, although Ismail Haniyeh, the prime minister of Hamas's government in the Gaza Strip, is a likely candidate.
Khaled Meshaal - Hamas' politburo chief living in exile - was declared the group's overall leader after Sheikh Yassin was killed in an Israeli missile strike.
Although attempts to forge a Palestinian government of national unity have since stalled, some observers believe Mr Meshaal could emerge as a candidate to head such an administration.
But the statement urged Mr Meshaal to leave it to the council to decide whether or not he should leave "on the basis of the higher interests of the movement".
Speaking to CNN's Wolf Blitzer in Cairo, Morsy said he invited Palestinian Authority President Mahmoud Abbas to meet with Khaled Meshaal, the political leader of Hamas, who lives in Cairo.
El acuerdo fue firmado por el presidente de la Autoridad Nacional Palestina, Mahmoud Abbas y el jefe de Hamas en el exilio, Khaled Meshaal, tras conversaciones llevadas a cabo en Meca, Arabia Saudita.
The Bush administration has urged former President Jimmy Carter not to go forward with plans to meet with Hamas leader Khaled Meshaal next week in Damascus, Syria, during a trip to the Mideast.
"Political bureau chief Khaled Meshaal has notified Hamas's consultative council that he does not wish to be a candidate for the movement's future leadership, " the AFP news agency quoted the statement as saying on Sunday.
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