Najib has been widerly expected to take over from Abdullah but the timing has not been announced.
Abdullah has been under pressure to resign from his post after the governing Barisan Nasional's dismal election performance in the general elections in March.
Barisan Nasional faced its worst results in 40 years, losing its two-thirds majority in the federal parliament and control of several states.
The elections were widely seen as a referendum on the rule of Abdullah.
Several key leaders in the United National Malays Organisation (Umno), the main partner in Barisan Nasional, advised Abdullah that he would lose if he stood for re-election as Umno's president later this year.
There are also been signs of possible strains between the Barisan Nasional-run federal government and the opposition-controlled states, including the reported shelving of major projects infrastructure projects in Penang and Selangor.
The federal government has denied claims that it is punishing voters for supporting the opposition and sabotaging development efforts by the new governments.
The announcement of a possible political transition came as more than 1,000 opposition supporters marched through Kuala Lumpur in the biggest public show of anger over the government's recent decision to raise fuel prices.
Riot police fanned out in Kuala Lumpur's commercial hub on Friday, setting up roadblocks and foiling the demonstrators' plan to walk from a popular mosque to the nearby Petronas Twin Towers, the headquarters of national oil company Petronas.
The protesters chanted "Down with Barisan Nasional" and called for Abdullah to resign.
It was the biggest in a series of protests that have plagued Abdullah's government since petrol shot up on June 5 by 41 per cent to 2.70 ringgit ($0.87) a litre, or 10.23 ringgit ($3.30) a gallon.
Opposition parties and public-rights groups plan to hold more protests in the next few weeks, culminating in what they say will be a rally by one million people in Kuala Lumpur on July 5.
Source: Al Jazeera and agencies