After a fast start to the game with a number of chances, mainly for the home side, the game settled into something of a stalemate.
That continued into the second period, although Rayan struck the post after a brilliant run from the Bournemouth half.
There were more chances for both sides late on as the game opened up, but neither side could break the deadlock, meaning the past six games in the league between these two sides have ended in a draw.
Andoni Iraola made two changes to his side which beat Everton, as Junior Kroupi and Tyler Adams started, the latter for his first appearance since returning from injury.
Ryan Christie and Alex Toth dropped to the bench as Marcos Senesi captained the side.
West Ham United's Axel Disasi attempts to score during the Premier League match at the London Stadium. (Image: John Walton/PA Wire)
The Hammers went close in the opening minutes as Axel Disasi flicked a corner towards goal, Djordje Petrovic saved well before Disasi shot over the bar from the rebound from close range.
West Ham threatened again in the early stages as Taty Castellanos delivered an inviting cross from the right, but Crysencio Summerville could only poke the ball straight at Petrovic.
A first shot on goal for the Cherries came with five minutes on the clock, as Rayan cut inside and fired straight at Mads Hermansen.
After that early flurry of opportunities, both sides had spells of pressure but without creating too much of a goal threat.
Castellanos headed over the bar from a corner, before Alex Scott powered straight at Hermansen from outside the box.
West Ham United's Taty Castellanos attempts a header on goal during the Premier League match at the London Stadium. (Image: John Walton/PA Wire)
Summerville then shot well over the bar after some neat footwork to get away from Senesi, as the sides entered the break level at 0-0.
Like the latter stages of the first period, changes were at a premium for both sides at the start of the second half.
The first shot on goal of the half went the way of the Hammers as Mateus Fernandes dragged a shot wide.
Rayan struck the post after a superb run from inside the Cherries half, as he drifted past defenders and shot from the edge of the box, but it clipped the upright and went behind shortly after the hour mark.
Bournemouth's Rayan shoots at goal during the Premier League match at the London Stadium. (Image: John Walton/PA Wire)
Iraola introduced David Brooks and Ryan Christie for Amine Adli and Adams with 24 minutes to play, and the Welshman’s first act was to see a shot blocked behind after a quick Cherries attack.
Castellanos shot wide with a bicycle kick before a Senesi mistake saw the Hammers come forward, but after James Hill cleared the ball off the toes of Fernandes, Soungoutou Magassa shot over.
An excellent dummy from Kroupi from a Brooks pass saw the ball come through to Rayan, but his shot was blocked.
Petrovic saved excellently from a Callum Wilson shot, and the former Cherry shot over the bar moments later.
Bournemouth's Alex Scott is tackled by West Ham United's Tomas Soucek during the Premier League match at the London Stadium. (Image: John Walton/PA Wire)
Scott saw an effort saved by Hermansen after neat build up play from the Cherries, before the Englishman deflected a Summerville effort onto the roof of the Cherries net.
Marcus Tavernier replaced Evanilson and made an impact, but after a driving run from the Englishman down the left wing, there was no one to meet his cut back.
Kroupi had a shot from range saved comfortably by Hermansen as Enes Unal and Adam Smith replaced the Frenchman and Alex Jimenez with one minute left of the 90.
Bowen blasted a shot high and wide after a quick West Ham counterattack, but that was the final chance of the game as it ended without score.
The Cherries move up to eighth in the table with 38 points, although Everton, who they move above, play on Monday night.
West Ham: 1; M. Hermansen; 12, E. Diouf; 4, A. Disasi; 15, K. Mavropanos; 29, A. Wan-Bissaka; 27, S. Magassa (Kante 84’); 18, M. Fernandes; 7, C. Summerville; 28, T. Soucek (booked, 33’)(Wilson 72’); 20, J. Bowen; 11, V. Castellanos.
AFC Bournemouth: 1, D. Petrovic; 3, A. Truffert; 5, M. Senesi (booked, 44’); 23, J. Hill; 20, A. Jimenez (Smith 89’); 12, T. Adams (Christie 66’); 8, A. Scott; 21, A. Adli (Brooks 66’); 22, J. Kroupi (Unal 89’); 37, Rayan; 9, Evanilson (Tavernier 80’).