Name for this pattern in woven fabric

Solution 1:

Your fabric is an example of a damask weave, probably in linen. It is difficult to tell from the image whether it is an ordinary (single) damask or a double damask.

The Wikipedia article linked to is not very helpful since it only shows furnishing damasks, which tend to be very heavy. The V&A website has plenty of examples of table linen but the images are not detailed since they are of entire pieces. Thomas Ferguson's website gives a much better impression of modern linen damasks.

Wikipedia: Damask

Victoria & Albert Museum: Linen damask napkin

Thomas Ferguson Irish Linen: Fine Irish linen damask

Solution 2:

I think you're thinking of damask. Your intuition that it's related to jacquard is on-point, since those looms can be used to make a "damask-look" material (see this post on The Dreamstress for a comparison of brocade, jacquard, and damask).

A general definition:

Damask (Arabic: دمسق‎‎) is a reversible figured fabric of silk, wool, linen, cotton, or synthetic fibres, with a pattern formed by weaving. Damasks are woven with one warp yarn and one weft yarn, usually with the pattern in warp-faced satin weave and the ground in weft-faced or sateen weave. (Wikipedia)

And from a textile-themed website article (emphasis mine):

[R]eal damask is actually a type of fabric which uses a variety of weaving techniques (most commonly satin and twill variants) to create areas of different sheen in the cloth. Because the different textures reflect light differently, the patterns show as variations in tone, and sometimes the weave is given even further emphasis by using different colours in the warp and weft. (Julie Gibbons, "History of Surface Design: Damask" PatternObserver.com)

Note that not all damasks are monochrome, and sometimes the term is used to describe patterns typical of some damask fabric (elaborate and scroll-y).

Some examples of damask table linens:

(And I just noticed that this has been answered in comments by @Mick while I was composing the answer; I'm going to post it anyway, in case folks want some additional references. I can make it a community wiki if that's the right etiquette.)