Where does the Constitution say the president may not suspend the Writ?
"Lincoln, with his usual incisiveness, put his finger on the debate that inevitably surrounds issues of civil liberties in wartime. If the country itself is in mortal danger, must we enforce every provision safeguarding individual liberties even though to do so will endanger the very government which is created by the Constitution? The question of whether only Congress may suspend it has never been authoritatively answered to this day, but the Lincoln administration proceeded to arrest and detain persons suspected of disloyal activities, including the mayor of Baltimore and the chief of police."
-- William Rehnquist, Chief Justice of the Supreme Court, November, 1999
Walt
The constitution does not indicate any executive power wrt the writ - thus none exists.