He made the announcement during the second day of his visit to the Egyptian capital, Cairo.
Saudi Arabia and other Gulf countries have supported Egypt with billions of dollars since President Sisi took power in 2013 following mass street protests.
Saudi Arabia regards Egypt as a crucial partner in efforts to build a bloc of friendly Sunni Muslim states as a bulwark against growing regional influence of Shia-led Iran.
King Salman`s visit comes amid recent strains in the relationship, with President Sisi taking a less hardline stance against Syrian President Bashar al-Assad and Riyadh seeking more support from Cairo for its war against rebels in Yemen.
Egyptian President Abdul Fattah al-Sisi said the bridge would be named after the Saudi king.
"I agreed with my brother his Excellency President Abdul Fattah al-Sisi to build a bridge connecting the two countries" the king said.
"This historic step to connect the two continents, Africa and Asia, is a qualitative transformation that will increase trade between the two continents to unprecedented levels" he added.
President Sisi said it marked "a new chapter on the road of Arab joint action".
A Red Sea bridge linking the two countries has been proposed several times before but has failed to become a reality.
Previous estimates for the bridge project suggested a cost of around $3-4bn (£2-3bn), but no further information has yet been released for the latest plan.
King Salman is currently on a five-day visit to Egypt, where he is expected to announce more trade and co-operation agreements.