My daughter is very shy and has difficulty in making friends at school. What can I do?

Friendships are a must for healthy development of a child but some children are perpetual lonely and the reasons could vary. 
There are many reasons why your daughter at school may lack friends. Extreme shyness and low self-esteem, high intelligence, poor social skills, notable differences between your daughter and her peer group, and learning disabilities can all make establishing friendships challenging.  Some children suffer from painful shyness. Even around children they know fairly well, they may hesitate to comment or participate. The fact that you are speaking to your child’s teacher is a great plus, as teachers can help her by praising your daughter when she does choose to participate. Play dates in a neutral setting with one other child. Alternately, your daughter may feel more confident on her home turf, and a play date in your home may offer a less crowded way to get to know other children.
Highly intelligent children may also have difficulty making friends. Their intelligence, especially in verbal language may make it difficult for other children to understand them. However, there is usually more than one child in a classroom with high intelligence. I think it’s a great idea by pairing two very verbal children together can be an excellent way of helping her establish friends. Teaching her social skills, like actually listening to another’s reactions, and responding appropriately, can also help her more easily make friends.
Its important having all the above in mind that no matter how strongly you feel about your daughter’s situation, if you try too much it could backfire. Friendships are not something that you can go out and buy your child like her favourite toy.
My best advice that you can do in situations like this is to be supportive, boost her confidence, be a good listener and help your daughter develop a sense of humour so that she can ride out difficult situations. 
I want you to consider that your daughter may not be lonely. She just may be a kind of person that needs more space and does not have a very strong need for companionship. There are many children who are quite happy curled up in a corner reading a book or engaged in other solitary activities. You should diagnose loneliness based on your observations of her interaction with her peers and listening to what she says. You should also keep an eye out for changes in behaviour like fitful sleep, moodiness or loss of appetite. If you are worried about your child's inability to make friends, you should continue to speak to her teacher or even to a counsellor if you feel the need. You should also keep in mind that some children have a greater need for social approval than others and their idea of no friends may just be that not everyone likes her. 

How you can continue to help

•  Invite children over to play especially younger children as playing with younger children can boost your daughter’s confidence. The older child tends to feel more in control and is less likely to be intimidated by a younger child.

•  Give your child advice about how she should go about making friends. Not all children are born with this ability so you may need to teach your child some social skills. Tell her to find friends with common interests and to show interest in other children's hobbies. 

•  If your child is upset about her lack of friends, do not try to gloss over the situation. Be sympathetic and vocalize the way she must be feeling.

•  When your child invites other children over, you can help break the ice by organizing a few games or activities initially before leaving them to their own devices. However, do not play the fussy mother hen all the time.

•  Observe her interactions with her peers and point out later where she might be going wrong. Maybe she's too bossy or not willing to share or too quick to take offence. Explain this to her in a way that doesn't get her back up so that she feels that you too are against her.

•  Encourage her to be persistent and take the effort. If another child has turned an invitation down, tell your child that there's no harm inviting her over again instead of taking the rebuff to heart and retiring hurt.

•  Do not encourage competition with her peers or engage in constant comparisons. This will foster feelings of insecurity and rivalry rather than friendship.

•  Do fun things with your child so that she gets the message that she is a good fun companion and not a bore as she may secretly think.

•  Enrol your child in hobby classes that will teach her a skill or highlight a talent or encourage her to participate in activities like team sport that will widen her social circle and boost her confidence.